New Worlds in Astroparticle Physics: Proceedings of the Fifth International Workshop, Faro, Portugal 8-10 January 2005

152

black hole this is expected to oscillate around the asymptotic Kerr value with the form of a quasi-normal mode. By fitting to this mode we extract an estimate of the angular momentum parameter a/Mhor as 10 -?—

= y/l-

(-1.55 + 2.55C,) 2 ,

(1)

where Mhor coincides with the black-hole mass M only if the spacetime has become axisymmetric and stationary. A third method for calculating instead J and hence measuring M is to use the isolated and dynamical-horizon frameworks of Ashtekar and collaborators 2,15 . This method assumes the existence of a Killing vector field intrinsic to a marginally trapped surface such as an apparent horizon and provides an unambiguous definition of the spin of the black hole and hence of its total mass. If there is an energy flux across the horizon, the isolated-horizon framework needs to be extended to the dynamical-horizon formalism3. The advantage of the dynamical-horizon framework is that it gives a measure of mass and angular momentum which is computed locally, without a global reconstruction of the spacetime. One possible disadvantage is that the horizon is required to have an (approximate) Killing vector field, so that no information can be found with this approach in the case in which the horizon deviates largely from axial symmetry. However, arbitrarily large distortions are allowed as long as they are axisymmetric in the sense of having a Killing vector field; the coordinate representation of the horizon does not need to have any symmetries. As expected, since our simulations do not deviate significantly from axisymmetry, we have not encountered problems in applying the dynamical horizon framework to the horizons found in our simulations. A fourth method for computing J only applies if an event horizon is found and if its angular velocity has been measured. In a Kerr background, in fact, the generators of the event horizon rotate with a constant angular velocity UJ = -gt^/gu — \Jgttl9u such that a

J

Aw2

V*

(2)

where A is the event-horizon proper area. The direct comparison of different methods employed allowed to measure the mass of the black hole with a precision of 0.5% in the case of model Dl and of ~ 1% in the case of model D4 a . a

T h e precision is based on the initial ADM mass computed on a compactified domain.

153 5. Reconstructing the global spacetime All of the results presented and discussed in the previous Sections describe only a small portion of the spacetime which has been solved during the collapse. It is also interesting and instructive to collect all of these pieces of information into a global description of the spacetime and look for those features which mark the difference between the collapse of slowly and rapidly rotating stellar models. As we discuss below, these features emerge in a very transparent way within a global view of the spacetime.

i

0.8

0.6

3 0.4 0.4

-

,

i

i

i

i

i

Js

L —-—-

i

'

'

)

...,.., 1 I

o.a

:

; J:^JJ ========£ 7^

4

F i g u r e 3. and D4.

i

-

0.6

0 4 B 12 polar and equatorial circumf. radii (km)

i

D4

0.2

n

i

.

1

1

1

I I

& 1

1

:

\

\

-

\!

•

.

1 1

0 4 8 12 polar and equatorial circumf. radii (km)

E v o l u t i o n of t h e m o s t r e l e v a n t surfaces d u r i n g t h e c o l l a p s e for t h e cases D l

To construct this view, we use the worldlines of the most representative surfaces during the collapse, namely those of the equatorial stellar surface, of the apparent horizon and of the event horizon. For all of them we need to use quantities measured by proper observers and, in particular, circumferential radii. The results of this spacetime reconstruction are shown in Fig. 3, whose left and right panels refer to the collapse of models Dl and D4, respectively. The different lines indicate the worldlines of the circumferential radius of the stellar surface (dotted line), as well as that of the apparent horizon (dashed line) and of the event horizon (solid line). Note that for the horizons we show both the equatorial and the polar circumferential radii, with the latter being always smaller than the former. For the stellar surface, on the other hand, we show the equatorial circumferential radius only. Note that in both panels of Fig. 3 the event horizon grows from an essentially zero size to its asymptotic value. In contrast, the apparent

154

horizon grows from an initially non-zero size and, as it should, is always contained within the event horizon. At late times, the worldlines merge to the precision at which we can compute them. A quick look at the two panels of Fig. 3 is sufficient to appreciate the different properties in the dynamics of the collapse of slowly and rapidly rotating models. The worldlines of the stellar equatorial circumferential radius are very different in the two cases. In the slowly rotating model Dl, in particular, the star collapses smoothly and the worldline always has a negative slope, thus reaching progressively smaller radii as the evolution proceeds (c/. left panel of Fig. 3). By time t ~ 0.59 ms, the stellar equatorial circumferential radius has shrunk below the corresponding value of the event horizon. In the case of the rapidly rotating model D4, on the other hand, this is no longer true and after an initial phase which is similar to the one described for Dl, the worldline does not reach smaller radii. Rather, the stellar surface slows its inward motion and, at around t ~ 0.6 ms, the stellar equatorial circumferential radius does not vary appreciably. Indeed, the right panel of Fig. 3 shows that at this stage the stellar surface moves to slightly larger radii. This behaviour marks the phase in which a flattened configuration has been produced and the material at the outer edge of the disc experiences a stall. As the collapse proceeds, however, also this material will not be able to sustain its orbital motion and, after t ~ 0.7 ms, the worldline moves to smaller radii again. By time t m 0.9 ms, the stellar equatorial circumferential radius has shrunk below the corresponding value of the event horizon.

6. Gravitational waves In addition to the technical difficulties due to the accurate treatment of the hydrodynamics involved in the collapse and reported above, the precise calculation of the gravitational radiation emitted in the process is particularly challenging as the energy released in gravitational waves is much smaller than the total rest-mass energy of the system. Indications of the difficulties inherent to the problem of calculating the gravitational-wave emission in rotating gravitational collapse have emerged in the first and only work, which dates back almost 20 years 28 . In 1985, in a landmark work in numerical relativity, Stark and Piran used their axisymmetric general-relativistic code to evolve rotating configurations and to compute the gravitational radiation produced by their collapse to black holes. The results referred to initial configurations consisting of polytropic stars which underwent col-

155 lapse after the pressure was reduced by a factor ranging from 60% up to 99% for the rapidly rotating models. The initial data effectively consisted of spherically-symmetric solutions with a uniform rotation simply "added" on. Not being stationary solutions of the Einstein equations, these stars could reach dimensionless spins up to a = J/M2 — 0.94, with J and M the angular momentum and mass of the star, respectively. Overall, their investigation revealed that while the nature of the collapse depended on the parameter a, the form of the waves remained roughly the same over the entire range of the values of a, with the amplitude increasing with a. Particularly important was evidence that, despite the complex matter dynamics during the collapse, the gravitational-wave emission could essentially be related to the oscillations of a perturbed black hole spacetime. The lack of published works about the gravitational-wave emission from 3D simulations of this process can be justified in part by the difficulties in measuring a signal often below the truncation error of the 3D simulations, but most importantly by the fact that all of the above calculations made use of Cartesian grids with uniform spacing. With the computational resources currently available, this choice, in fact, places the outer boundaries too close to the source to detect gravitational radiation. The progressive mesh-refinement (PMR) techniques we have used have removed these restrictions, enabling us to place the outer boundaries of the computational domain at very large distances from the collapsing star. This conceptually simple but technologically challenging improvement has two important physical consequences. Firstly, it reduces the influence of inaccurate boundary conditions at the outer boundaries of the domain whilst retaining the required accuracy in the region where the black hole forms. Secondly, it allows the wave-zone to be included in the computational domain and thus to extract the gravitational waves produced in the collapse. In practice, we have adopted a Berger-Oliger prescription for the refinement of meshes on different levels9 and used the numerical infrastructure described in Schnetter et al. 25 . In addition to this, we have also implemented a simplified form of adaptivity in which new refined levels are added at predefined positions during the evolution. More specifically, given an initial stellar model of mass M and equatorial coordinate radius R, our initial grid consists of four levels of refinement, with the innermost one covering the star with a typical resolution of Ax ~ 0.17 M and with the outermost having a typical resolution of Ax ~ 1.38 M and extending up to ~ 82.5 M ~ 20.9 R. As the collapse proceeds and the star occupies smaller portions of the computational domain, three more refined levels are

156

added one by one, nested in the four original ones. By the time the simulation is terminated at ~ 81.5 M, the finest typical spatial resolution is Aa; ~ 0.02 M. A detailed discussion of the grid and of its evolution will be given in Baiotti et al. 8 . Also, hereafter we will restrict the discussion to the collapse of the most rapidly rotating dynamically unstable model, namely model D4. The discussion of the emission from stellar models rotating at smaller velocities will be presented in Baiotti et al. 8 . For the gravitational-wave extraction, we have adopted a gaugeinvariant approach in which the spacetime is matched with the nonspherical perturbations of a Schwarzschild black hole 1 ' 24,13,23 ' 4 ' 17 . In practice, a set of "observers" is placed on 2-spheres of fixed coordinate radius r e x , where they extract the gauge-invariant, odd Q^ and even-parity $ ^ metric perturbations 21 . Here £, m are the indices of the angular decomposition and we usually compute modes up to I = 5 with m = 0; modes with m ^ 0 are essentially zero because of the high degree of axisymmetry in the collapse. Although the position of such observers is arbitrary and the information they record must be the same for waves extracted in the wavezone, we place our observers between 40 M and 50 M from the centre of the grid so as to maximize the length of the extracted waveform. We note that while a similar choice was made by Stark and Piran 28 , it still provides only an approximate description of what would be observed at spatial infinity.

0.1

t-r (ms) 0.2

0.3

0.02

0.1

t-r (ms) 0.2 0.3

0.002 0.001

-0.01 -0.003 -0.02 10

20 30 t-r (M/M8)

10

20 30 t-r (M/M0)

Figure 4. LEFT: The £ = 2, even-parity perturbation as extracted by observers at different positions r e x expressed in retarded time. RIGHT: The ( = 4 mode, with the inset offering a comparison in the amplitudes of the two modes.

Using the odd and even-parity perturbations Qfm = XQ^l and Q^m =

157 A * ^ , where A = y/2{£ + 2)\/{£ - 2)!, we report in Fig. 4 and for model D4 the lowest-order multipoles for Qjm with the offset produced by the stellar quadrupole removed8. The left panel, in particular, refers to the £ = 2 mode as extracted by four different observers at increasing distances and expressed in retarded time. The right panel instead refers to the £ = 4 mode, with the inset giving a comparison between the two modes and showing that the gravitational-wave signal is essentially quadrupolar, with the £ = 2 mode being about an order of magnitude larger than the £ = 4 mode. The very good overlap of the waveforms measured at different positions is important evidence that the extraction has been performed in the wavezone, since the invariance under a retarded-time scaling is a property of the solutions of a wave equation. The overlap disappears if the outer boundary is too close or when the waves are extracted at smaller radii. A similar overlap is seen also for the £ = 4 mode (not shown). Another indication that the waveforms in Fig. 4 are an accurate description of the gravitational radiation produced by the collapse comes by analysing their power spectra. The collapse can be viewed as the rapid transition between the spacetime of the initial equilibrium star and the spacetime of the produced rotating black hole. It is natural to expect that the waveforms produced in this process will reflect the basic properties of both spacetimes and in particular the fundamental frequencies of oscillation. We validate this in Fig. 5 (left), where we show the power spectral densities (PSD) of the waveforms of the metric perturbations Q^Q and Q^0 reported in Fig. 4 (the units on the y-axis are arbitrary). The upper panel of Fig. 5 (left), in particular, shows the PSD of Q j 0 and compares it with the frequencies of the £ = 2,m = 0 quasi-normal mode (QNM) of a Kerr black hole with M = 1.861 M Q and a = 0.620 (dashed line at 6.7 kHz) as well as with the first wu "interface" mode for a typical compact star with M = 1.27 MQ and R = 8.86 km (dotted line at 8.8 kHz) 19 . Similarly, the lower panel of Fig. 5 (left), shows the PSD of Q j 0 comparing it with the £ = 4, m = 0 QNM of a Schwarzschild black hole 19 (dashed line at 14.0 kHz) and the first w\ "curvature" mode 19 (dotted line at 12.8 kHz). The frequencies for the QNMs of the black hole and for the tu-modes of the star are those most easily obtained from tabulated values in the literature and serve here as indicative references. It is apparent that both peaks in the PSDs are rather far from the maximum sensitivity area of modern interferometric and bar detectors.

158

' ' ' I ' ' ' ' I '

(r/M)h +

8= n / 2

-0.002 r|

i : ,i f-1 i •! i j i i i | i i i | i i i

-I

0.00015

1-4 QNM of t h e bh

I I I [ I I I I | I I I I | I I I I | I I

" (r/M)h

if, QNM of typ. s t a r -

r M /M = 44.1 J 0.00015 10

20

30 40 f (kHz)

50

60

10

20

30

t - r (M/MJ

Figure 5. LEFT: Power spectra of the waveforms reported in Fig. 4. The dashed vertical lines indicate the frequencies of the QNMs of a black hole, while the dotted ones the W\ and wu modes of a typical star. RIGHT: gravitational-wave amplitudes in the T T gauge.

Although the waveforms have very short duration with very broad PSDs, Fig. 5 (left) shows that these have strong and narrow peaks (a similar behaviour can be shown to be present also for the odd-parity modes 8 ). Indeed, the excellent agreement between the position of these peaks and the fundamental frequencies of the vacuum and non-vacuum spacetimes is an important confirmation of the robustness of the results obtained. Using the extracted gauge-invariant quantities it is also possible to calculate the transverse traceless (TT) gravitational-wave amplitudes in the two polarizations h+ and hx as h

+ ~ ih» = I £ (QL - i /Wm(*')d*') -,Y'm,

(3)

where _2Ytm is the s = — 2 spin-weighted spherical harmonic. Because of the small amplitude of higher-order modes, the TT wave amplitudes can be simply expressed as h+ ~ /i+(Q20>Q4o) an( ^ ^x — hxiQmiQw), where Q30 ^> Q£Q- Their waveforms are shown in Fig. 5 (right) for the detector at r e x = 37.1 M and for two different inclination angles 8 . Note that the amplitudes in the cross polarization are about one order of magnitude smaller than those in the plus polarization, with the maximum amplitudes in a ratio |(r/M)/ix|max/|(r/M)/i + | m a x ~ 0.06. Here the odd-parity perturbations, which are zero in spacetimes with axial and equatorial symmetries, are just the result of the coupling, induced by the rotation, with the even-parity perturbations.

159 A precise comparison of the amplitudes in Fig. 5 (right) with the corresponding ones calculated by Stark and Piran 28 is made difficult by the differences in the choice of initial data and, in particular, by the impossibility of reaching a > 0.54 when modelling consistently stationary polytropes in uniform rotation. However, when interpolating their results for the relevant values of a, we find a very good agreement in the form of the waves, but also that our estimates are about one order of magnitude smaller, with | ( r / M ) / i + | m a x ^ 0.00225. Furthermore, we observe the amplitude of the gravitational waves to increase with the pressure reduction, suggesting that the origin of the difference is related mainly to this (see Baiotti et al. 8 ). Following Thorne 31 and considering the optimal sensitivity of VIRGO for the burst signal only, we set an upper limit for the characteristic amplitude produced in the collapse of a rapidly and uniformly rotating polytropic star at 10 kpc to be hc = 5.77 x 10~22(M/M&) at a characteristic frequency fc = 931 Hz. In the case of LIGO I, instead, we obtain hc = 5.46 x 1 0 - 2 2 ( M / M e ) at fc = 531 Hz. In both cases, the signalto-noise ratio is S/N ~ 0.25, but this can grow to be < 4 in the case of LIGO II. These ratios could be increased considerably with the detection of the black hole ringing following the initial burst (see Baiotti et al. 8 ). Computing the emitted power as

e,m \

/

the total energy lost to gravitational radiation is E = 1.45 x 10~ 6 (M/M©). This is about two orders of magnitude smaller than the estimate made by Stark and Piran 28 for a star with a = 0.54, but still larger than the energy losses computed recently in the collapse of rotating stellar cores to protoneutron stars by Miiller et al. 22 .

7. Conclusion We have developed recently developed and tested Whisky, a 3D generalrelativistic numerical evolution code that combines state-of-the-art numerical methods for the spacetime evolution with an accurate hydrodynamical evolution employing several high-order HRSC methods and hydrodynamical excision techniques. The evolution of the spacetime and of the hydrodynamics is coupled transparently through the method of lines, which allows for the straightforward implementation of various different time-integrators.

160 As a first astrophysical problem for this novel setup, we have here focused on the collapse of rapidly rotating relativistic stars to Kerr black holes. The stars are assumed to be in uniform rotation and dynamically unstable to axisymmetric perturbations. While the collapse of slowly rotating initial models proceeds with the matter remaining nearly uniformly rotating, the dynamics is shown to be very different in the case of initial models rotating near the mass-shedding limit. Although the stars become highly flattened during collapse, attaining a disc-like shape, the collapse cannot be prevented because the specific angular momentum is not sufficient for a stable disc to form. Instead, the matter in the disc spirals towards the black hole and angular momentum is transferred inward to produce a spinning black hole. Several different approaches have been employed to compute the mass and angular momentum of the newly formed Kerr black hole. Besides more traditional methods involving the measure of the geometrical properties of the apparent and event horizons, we have fitted the oscillations of the perturbed Kerr black hole to specific quasi-normal modes obtained by linear perturbation theory. In addition, we have also considered the recently proposed isolated and dynamical horizon frameworks, finding it to be simple to implement and yielding estimates which are accurate and more robust than those of other methods. This variety of approaches has allowed for the determination of both the mass and angular momentum of the black hole with an accuracy unprecedented for a 3D simulation. We have also presented the first waveforms from the gravitational collapse of rapidly rotating stars to black holes using 3D grids with Cartesian coordinates. The great potential shown by the PMR techniques employed here opens the way to a number of applications that would be otherwise intractable with uniform Cartesian grids. Work is in progress to consider initial models with realistic EOSs or in differential rotation, for which values o > 1 can be reached and more intense gravitational radiation is expected.

Acknowledgments It is a pleasure to thank T. Font, F. Loffler, P. Montero, E. Seidel, N. Stergioulas and O. Zanotti for useful discussions. We are also grateful to the Cactus-code team, for their efforts in producing an efficient infrastructure for numerical relativity. The computations were performed on the Albert 100 cluster at the University of Parma and on the Peyote cluster at the AlbertEinstein-Institut.

161

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

29. 30. 31.

G. Allen, K. Camarda, E. Seidel, gr-qc/9806036 (1998) A. Ashtekar, C. Beetle, S. Fairhurst, Class. Quantum Grav. 17, 253 (2000) A. Ashtekar, B. Krishnan, Phys. Rev. Lett. 89, 261101 (2002) J. Baker, S. Brandt, M. Campanelli, C. O. Lousto, E. Seidel, R. Takahashi, Phys. Rev. D 62, 127701 (2000) L. Baiotti and I. Hawke and P. Montero and L. Rezzolla, Mem. Soc. Astron. It., 1, 327 (2003) L. Baiotti, I. Hawke, P. J. Montero, F. Loftier, L. Rezzolla, N. Stergioulas, J.A. Font, E. Seidel, Phys. Rev. D 7 1 , 024035 (2005) L. Baiotti, I. Hawke, L. Rezzolla, E. Schnetter, Phys. Rev. Lett., 94, in press, (2005) L. Baiotti et al., in preparation (2005) M.J. Berger, J. Oliger, J. Comput. Phys. 53, 484 (1984) S. Brandt, E. Seidel, Phys. Rev. D52, 856 (1995) S. Brandt J. A. Font, J. M. Ibanez, J. Masso, E. Seidel, Comp. Phys. Comm. 124, 169 (2000) http://www.cactuscode.org K. Camarda, E. Seidel, Phys. Rev. D 59, 064019 (1999) P. Diener, Class. Quantum Grav. 20, 4901,(2003) O. Dreyer, B. Krishnan, D. Shoemaker, E. Schnetter, Phys. Rev. D67, 024018 (2002) M.D. Duez, S.L. Shapiro, H.J. Yo, Phys. Rev. D 69, 104016 (2004) J.A. Font, T. Goodale, S. Iyer, M. Miller, L. Rezzolla, E. Seidel, N. Stergioulas, W. M. Suen, M. Tobias, Phys. Rev. D 65, 084024 (2002) I. Hawke, F. Loffler, A. Nerozzi, gr-qc/0501054. K.D. Kokkotas, B.G. Schmidt, Living Rev. Relativity 2 (1999) E.W. Leaver, Proc. R. Soc. Lond. A 402, 285 (1985) V. Moncrief, Annals of Physics 88, 323 (1974) E. Miiller, M. Rampp, R. Buras, H.T. Janka, D.H. Shoemaker, Astrophys. J. 603, 221 (2004) L. Rezzolla, A. M. Abrahams, R. A. Matzner, M. Rupright, S. L. Shapiro Phys. Rev. D 59, 064001 (1999) M.E. Rupright, A.M. Abrahams, L. Rezzolla, Phys. Rev. D 58, 044005 (1998) E. Schnetter, S.H. Hawley, I. Hawke, Class. Quantum Grav. 21, 1465 (2004) M. Shibata, T.W. Baumgarte, S.L. Shapiro, Phys. Rev. D 61, 044012 (2000) M. Shibata, Phys. Rev. D 67, 024033 (2003); Astrophys. J. 595, 992 (2003) R.F. Stark, T. Piran, Phys. Rev. Lett. 55, 891 (1985); Proceedings of the workshop Dynamical spacetimes and numerical relativity, Philadelphia, CUP, Cambridge UK, pp. 40-73 (1986) N. Stergioulas, J. L. Friedman, Astrophys. J. 444, 306 (1995). J. Thornburg,Class. Quantum Grav.,21, 743 (2004) K. S. Thorne in 300 Years of Gravitation, S. W. Hawking, W. Israel, eds., CUP, Cambridge, UK (1987)

T H E ROLE OF D I F F E R E N T I A L ROTATION IN T H E EVOLUTION OF T H E r - M O D E INSTABILITY

P A U L O M. SA A N D B R I G I T T E T O M E Departamento de Fisica and Centro Multidisciplinar de Astrofisica CENTRA, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal E-mail: [email protected], [email protected]

We discuss the role of differential rotation in the evolution of the I — 2 r-mode instability of a newly born, hot, rapidly-rotating neutron star. It is shown that the amplitude of the r-mode saturates in a natural way at a value that depends on the amount of differential rotation at the time the instability becomes active. It is also shown that, independently of the saturation amplitude of the mode, the star spins down to a rotation rate that is comparable to the inferred initial rotation rates of the fastest pulsars associated with supernova remnants.

1. Introduction The r-modes are non-radial pulsations modes of rotating stars that have the Coriolis force as their restoring force and a characteristic frequency comparable to the rotation speed of the star 1 . These modes are driven unstable by gravitational radiation in all rotating stars 2 . A deeper understanding of r-modes and their astrophysical relevance requires taking into account nonlinear effects in the evolution of the r-mode instability. One such a nonlinear effect, differential rotation induced by r-modes, has been studied by several authors. Rezzolla, Lamb and Shapiro 3 were the first to suggest that differential rotation drifts of kinematical nature could be induced by r-modes, deriving an approximate analytical expression for this differential rotation. Afterwards, the existence of such drifts was confirmed numerically both in general relativistic 4 and Newtonian hydrodynamics 5 . Recently, an exact r-mode solution, describing differential rotation of pure kinematic nature that produces large scale drifts along stellar latitudes, was found within the nonlinear theory up to second order in the mode's amplitude 6 . In this paper we discuss the role of differential rotation in the evolution of the / = 2 r-mode instability of a newly born, hot, rapidly-rotating neutron star.

162

163 2. The evolution model of Owen et al. A few years ago, Owen et al.7 proposed a simple model to study the evolution of the r-mode instability in newly born, hot, rapidly-rotating neutron stars. Within this model, it is assumed that the time evolution of the system (star and r-mode perturbation) is characterized by two parameters: the angular velocity of the star, fi(£), and the amplitude of the mode, a(t). The total angular momentum of the star is then given by J = IQ + JC

(1)

where the momentum of inertia / of the equilibrium configuration is / = I MR2, with / = 0.261, and the canonical angular momentum Jc of the rmode perturbation is Jc = - 3 / 2 a 2 f i J M i ? 2 , with J = 1.635 x 1CT2. Here, only the / = 2 r-mode is being considered and it is assumed that the mass density p and the pressure p of the fluid are related by a polytropic equation of state p = kp2, with k such that the mass and radius of the star take the values M = 1 . 4 M Q and R = 12.53 km, respectively. Within this model it is also assumed that the total angular momentum of the star decreases due to the emission of gravitational radiation and that the physical energy of the r-mode perturbation (in the co-rotating frame) increases due to the emission of gravitational radiation and decreases due to the dissipative effect of viscosity. These assumptions lead then to a system of differential equations for the angular momentum of the star, fi, and the amplitude of the r-mode, a: dQ ~dt da ~dt

_ 2Q Qa2 " " r ^ l + Qa2' _ a a 1 — Qa2 ~ ~ ^ ~ ^ l + Qa2'

(

' ( )

where Q = 3J/(27) = 0.094, the gravitational-radiation and viscous timescales, TQR and Ty, are given by 8

J_

=

TGR

J_

J_ ( A2 TGR

=

(4)

\nGp

J_/10^K\ 2

rv ~ fs \

T

)

1 / T \ 6 / Q2 +

fB V10 9 KJ

\nGp

(5)

In the above expressions, the fiducial timescales are ?GR = —3.26 s, fs = 2.52 x 108 s and fB = 6.99 x 108 s. For a newly born, hot, rapidly-rotating neutron star there is an interval of relevant temperatures and angular velocities of the star for which

164 the gravitational timescale is much smaller than the viscous timescale, TGR ^ TV- Therefore, for this interval of temperatures and angular velocities, we can neglect in the right-hand side of Eqs. (2) and (3) the terms proportional to Ty1 and obtain the solution fi = fio and a = aoexp{ — (t — to)/TOR}If the initial angular velocity is chosen to be fio = &K, where D,K = (2/3)\/7rGp is the Keplerian angular velocity at which the star starts shedding mass at the equator, then TQR = —37.1s, implying that the perturbation grows exponentially from an initial amplitude ao = 1 0 - 6 to values of order unity in just about 500 s 7 . After this first stage of evolution of the r-mode instability, the amplitude a has to be forced, by hand, to take a certain saturation value a3at ^ Q"1^2 = 3.26; in the second stage of the evolution it is then assumed that a = aaat and Q. is determined from the equation dtt dt

=

2Q alatQ 2 TGRl-a satQ'

W

which yields the solution [1-Qa2sat

\QKJ

\ sec J

J

where i* is the time at which occurs the transition from the first to the second stage of evolution. During the second stage of evolution, the star loses its angular momentum, spinning down to its final angular velocity. As can be seen from the above equations, the final angular velocity of the star depends critically on the saturation value of the mode's amplitude asat\ for instance, after one year of evolution, for CIQ = Q.K and asat = 1 one obtains Cl ~ O.lfi^, while for asat = 1 0 - 3 the angular velocity is fi ~ 0.9f2if. The fact that the growth of the mode's amplitude has to be stopped by hand at a certain saturation value introduces an element of arbitrariness into the solution, permitting, for instance, that agreement between the predicted final value of the angular velocity of the star and the value inferred from astronomical observations can always be achieved by fine-tuning the value of the saturation amplitude. As will be seen in the next section, this arbitrariness disappears when differential rotation is taken into account. 3. Evolution model with differential rotation Differential rotation induced by r-modes does contribute to the physical angular momentum of the r-mode perturbation 6 . Therefore, a model for

165

the evolution of the r-mode instability should include the effect of differential rotation. Here, we adopt the model of Owen et al.7, described in the previous section, with the important difference* that the total angular momentum of the star is given by j = m + swj,

(8)

where the physical angular momentum of the r-mode perturbation S^J is 6 6™J = i Q 2 n ( 4 K + b)JMR2.

(9)

In the previous expression, K is a constant fixed by the choice of initial data and gives the initial amount of differential rotation associated with the r-mode. The specific case K = —2, for which the physical angular momentum of the r-mode perturbation coincides with the canonical angular momentum, corresponds to the model of Owen et al.7 described in great detail in the previous section. There is not, to our knowledge, any physical condition that forces K to take such a particular value K = —2. Therefore, we will consider in this paper arbitrary values of K in the interval —5/4 ^ K < 10 13 b . Following the procedure described in the previous section, we arrive at a system of two first-order coupled differential equations determining the time evolution of the amplitude of the r-mode a(t) and of the angular velocity of the star Q,{t): — = -(K + 2)Q — , at 6 TQR da l + ^(K + 2)Qa2 ~dt

(10)

-2L,

(ID

TGR

valid when the damping effect of viscosity can be neglected relatively to the driving effect of gravitational radiation. a

T h e r e is another difference, albeit less significant, between the model of Owen et al.7 and the model we adopt here, namely, we deduce the evolution equations just from angular momentum considerations. b T h e upper limit for K results from the fact that one wishes to impose the condition that \5(2)j(to)\
166

This system of equations was solved analytically and discussed in great detail in Ref. [9]. In the initial stages of the evolution of the r-mode instability a increases exponentially 9 ,

Oo_y

a(t) ~ a0 exp { 0.027

t-V

(12)

sec while for later times, a increases very slowly as 3/5

2 48

/ .

. v 1/10

1

t-t0 sec

"<" = - l £

(13)

The amplitude of the mode saturates in a natural way and a smooth transition between the regimes (12) and (13) occurs for £ —£0 — fewx 102 seconds (see Fig. 1). This contrasts with the model of Owen et al.7, in which, after the short initial period of exponential.growth, the amplitude a has to be forced by hand to take a certain saturation value asat ^ Q~ 1|/2 . As can 10

-1

1

1

'

1

1

1

S/4

1 0.1

_

/""lO

103

yT

-

105

/

0.01

10 7

10' 3 .

-

10""

-

10'= io-<

/

.

1

200

.

1

.

1

400 600 (t-tg)/sec

i

800

1000

Figure 1. Time evolution of the amplitude of the r-mode for different values of K. The initial values of the amplitude of the mode and of the angular velocity of the star are, respectively, «o = 1 0 - 6 and Ho = SIK-

be seen from Eq. (13), the saturation value of the amplitude of the mode depends crucially on the parameter K, namely, asat oc {K + 2)~1/2. If K cz 0, corresponding to a situation in which the initial amount of differential rotation is small, then the r-mode saturates at values of order unity. If, on the other hand, the initial differential rotation associated to r-modes is significant, then the saturation amplitude a3at can be as small as 10~ 3 - 10~ 4 .

167 Let us now turn our attention to the time evolution of the angular velocity of the star, fi. In the initial stages of the evolution of the r-mode instability 0 decreases as 9

t-to"

f s i-^ +I ^U(^

(14)

sec

while for later times fi decreases slowly as -6/5

t-to

-1/5

(15)

sec The smooth transition between the regimes (14) and (15) occurs for t — to ~ few x 102 seconds (see Fig. 2). Remarkably, in the later phase of the

400

600

1000

(t-y/sec Figure 2. Time evolution of the angular velocity of the star for different values of K. The initial values of the amplitude of the mode and of the angular velocity of the star are, respectively, ao — 10~ 6 and Clo = ^IK-

evolution, the angular velocity fi does not depend on the value of K and, consequently, does not depend on the saturation value of a. This contrasts with the results obtained in Ref. [7], where the value of fi depends critically on the choice of asat. After about one year of evolution, when the dissipative effect of viscosity becomes dominant and starts damping the mode, the angular velocity of the star becomes Clone y e a r — 0.042O/<- (forfio= &K), in good agreement with the inferred initial angular velocity of the fastest pulsars associated with supernova remnants.

168 4. Conclusions In this paper we have discussed the role of differential rotation in the evolution of the I = 2 r-mode instability of a newly born, hot, rapidly-rotating neutron star. We have shown that, a few hundred seconds after the mode instability sets in, the amplitude of the r-mode saturates in a natural way at values that depend on the initial amount of differential rotation associated to the r-mode. If the initial differential rotation of r-modes is small, then the r-mode saturates at values of order unity. On the other hand, if the initial differential rotation is significant, then the saturation amplitude can be as small as 1 0 - 3 —10 - 4 . These low values for the saturation amplitude of r-modes are of the same order of magnitude as the ones obtained in recent investigations on wind-up of magnetic fields3 and on nonlinear mode-mode interaction 10 . We have also shown that the value of the angular velocity of the star becomes, after a short period of evolution of the r-mode instability, very insensitive to the saturation value of the mode's amplitude. After about one year of evolution the angular velocity is 0.042Q^- (for any asat), in good agreement with the inferred initial angular velocity of the fastest pulsars associated with supernova remnants. Acknowledgments We thank Oscar Dias and Luciano Rezzolla for helpful discussions. This work was supported in part by the Fundagao para a Ciencia e a Tecnologia (FCT), Portugal. BT acknowledges financial support from FCT through grant PRAXIS XXI/BD/21256/99. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

J. Papaloizou and J. E. Pringle, Mon. Not. R. Astron. Soc. 182, 423 (1978). N. Andersson, Astrophys. J. 502, 708 (1998). L. Rezzolla, F. K. Lamb and S. L. Shapiro, Astrophys. J. 531, L139 (2000). N. Stergioulas and J. A. Font, Phys. Rev. Lett. 86, 1148 (2001). L. Lindblom, J. E. Tohline and M. Vallisneri, Phys. Rev. Lett. 86, 1152 (2001). P. M. Sa, Phys. Rev. D. 69, 084001 (2004). B. J. Owen, L. Lindblom, C. Cutler, B. F. Schutz, A. Vecchio and N. Andersson, Phys. Rev. D. 58, 084020 (1998). L. Lindblom, B. J. Owen and S. M. Morsink, Phys. Rev. Lett. 80, 4843 (1998). P. M. Sa and B. Tome, Phys. Rev. D 71, 044007 (2005). P. Arras, E. E. Flanagan, S. M. Morsink, A. K. Schenk, S. A. Teukolsky and I. Wasserman, Astrophys. J. 591, 1129 (2003).

ANALYTICAL r-MODE SOLUTION WITH GRAVITATIONAL RADIATION REACTION FORCE

OSCAR J. C. DIAS* Perimeter Institute for Theoretical Physics, 31 Caroline St. N., Waterloo, Ontario N2L 2Y5, Canada E-mail: odias@perimeterinstitute. ca PAULO M. SA Departamento de Fisica and Centro Multidisciplinar de Astrofisica - CENTRA, Universidade do Algarve, Campus de Gambelas, 8005-139 Faro, Portugal E-mail: [email protected]

We present and discuss the analytical r-mode solution to the linearized hydrodynamic equations of a slowly rotating, Newtonian, barotropic, non-magnetized, perfect-fluid star in which the gravitational radiation reaction force is present.

1. I n t r o d u c t i o n T h e r-modes, which are pulsation modes of rotating stars t h a t have the Coriolis force as their restoring force, are driven unstable by gravitational radiation ( G R ) 1 . In the frame co-rotating with the star, the energy of the unstable r-mode grows exponentially, E = Eae~2t/TaR, with the gravitational timescale TQR given b y 2 , 3

where G is the Newton's constant, c is the velocity of light, Q is the angular velocity of the star and J = J0 drpr6 with p and R being, respectively, the density and the surface's radius of the u n p e r t u r b e d star. For a wide range of relevant temperatures and angular velocities of newly born, hot, rapidlyrotating stars, bulk and shear viscosity do not suppress the exponential growth of the energy of the r - m o d e 2 , 3 . *also at Centro Multidisciplinar de Astrofisica - CENTRA.

169

170

In the above-mentioned investigations 2,3 , the linearized hydrodynamics equations with the GR force were used to obtain an expression for the time evolution of the physical energy of the r-mode perturbation, dE/dt, from which the gravitational radiation and viscous timescales were determined. In this work, in which the main results of Ref. [4] are reported, we present an explicit expression for the r-mode velocity perturbations that solves the linearized hydrodynamics equations with the GR force. Our conclusions are that (i) these velocity perturbations are sinusoidal with the same frequency as the well-known GR force-free linear r-mode solution, (ii) the GR force drives the r-modes unstable with a growth timescale that agrees with the expression found in Refs. [2,3] and (iii) the amplitude of these velocity perturbations is corrected, relatively to the GR force-free case, by a term of order Q6.

2. Hydrodynamic equations with GR reaction force The Newtonian hydrodynamic equations for a uniformly rotating, barotropic, non-magnetized, perfect-fluid star in the presence of the gravitational radiation (GR) reaction force are the Euler, continuity, and Poisson equations given, respectively, by dtv+(v-

V)iT=-p-1VP-V$ + FGR, dtp + V-(pff) = 0, 2

V $

= 4TTGP,

(2) (3) (4)

where p, P and v are, respectively, the density, the pressure and the velocity of the fluid, $ is the Newtonian potential, and the GR reaction force,

FGR = -dt0+vx(VxP),

(5)

is assumed to be given by the 3.5 post-Newtonian order expansion that includes the contribution of the current quadrupole moment, which is the main responsible for the GR instability that sets in 5 ' 6 . In Eq. (5), /? is the gravitational vector potencial whose components are given by ^ 45c^ eiJkXixQ

kqi

W

where Sa(t) is the time-varying current quadrupole tensor, Sij(t)=

/ d3xekq{iXj)XkPvq,

(7)

171 Cijk is the Levi-Civita tensor, i 4 is the Cartesian coordinate of the point at which the tensor is evaluated, and S\j(t) denotes the nth time derivative of Sij.

3. Linearized hydrodynamics equations with GR reaction force The hydrodynamic equations (2)-(4) can be linearized, yielding dtS^v

+ (S^v- V)tf + (tf • V ) J ( 1 V = -VS^U

+ dWPGK,

dtSWp + $ • VSWp + V • (pS^v) = 0, 2

1

V ^ ^ = AnG6^p,

(8) (9) (10)

where v = fir sin Oe^ is the velocity of the unperturbed star, p its mass density, 8^Q denotes the first-order Eulerian change in a quantity Q and we have defined c5(1)t/ = S^P/p + (5(1)$. To compute the first-order Eulerian change in the GR force, Si-1^FGR, we assume that the GR force-free r-mode velocity perturbations", 6t%r = 0, S^he = -iJ^^-r2 S ^

sin 0ei^+u't\

2

i

= iy|^r sin(9cos6le W+

(H) u;t

),

act as a source for the first-order Eulerian change in the current quadrupole tensor, S^Sij. In the above expression, a is the amplitude of the r-mode and we assume that w = wo + iw, where the frequency of the r-mode, u>o = part that is related to the growth timescale w = Im[w] < 0, are arbitrary parameters to Under the above assumptions, 5^SXX is

(12) Re[u>], and the small imaginary of the instability of the mode, be determined. computed to be

<5(1)Sx* = - a f i * / J i e- ro V Wot ,

(13)

V 5 it where we have neglected the contribution coming from the terms iiiS^p (of order afi 3 ) and retained only the dominant terms p5^Vi (of order afl). a

I n this article, we will be concerned exclusively with the I — 2 r-mode, which is the most unstable mode.

172

Similarly, it is straightforward to show that the first-order Eulerian change in the other components of the quadrupole tensor satisfy the relations yy

—

"Jiij

6™SXZ = 6WSV2 = 5^SZZ = 0.

(14)

Using Eqs. (13) and (14), the first-order Eulerian change in the gravitational vector potential, 8^(3, is then computed to be SM0r = 0, SWfy = - K j | a f i ^ r 2 s i n l 9 e - r o t e w + a ' ° * \ 6(l)j34> = -injictttu>%r'2sm6cosee-™tei(-2't'+uot\

(15)

where the constant K sets the strength of the GR reaction force and is defined as

*S45V5?-

(16)

Finally, taking into account that Sy = 0, implying that fa = 0, the first-order Eulerian change in the GR force, 6^FGR, is computed to be j(i) F GR

=

J(l)/TGR

=

5(DFGR

_3iK^an2c^r2sin20cos0e-roV(2'*,+w°t\ j K ^ a Q ^ S ^Q

+

^

gin2

^

r2 s i n

2

, t i

Q g-ortgiCa^+wot) ) +

(17)

t

-Kj[aQ,w%r sin6cos6e- * e V't' "° \

=

4. The analytical r-mode solution with GR reaction force The linearized hydrodynamic equations (8)-(10), with the first-order Eulerian change in the GR force, 6^FGR, given by Eq. (17), admit the solution4 d^Vr = 0, 6^v9 = - i2 Va f t

f,/fi-H7(A-l)f25

# % = \aQ, hy/ZJi

+h(A-

r2sm6e-atei^+OJot\

l)fi5

(ig)

r2sm9cos6e-u>tei<-2't>+u'ot\

and 6P>U = laQ2 3

(LflL+iiAsA

\2\TTR

I

r 3 s i n 2 ^ c o s ^ e - r o V ^ + ^ t ) , (19)

with xv and LJQ given by w = -^J^jRU6

and

w0 = - ^ -

(20)

173 The velocity perturbations given by Eq. (18) have a piece similar to the GR force-free solution, the difference being the factor e _ r o t responsible for the exponential growth of the r-mode amplitude due to the presence of a GR reaction force, and another piece proportional to cey{A — l)fi 6 , where A is a constant fixed by the choice of initial data and 7 = 1024/tJ/(81fi). The GR force-free limit is obtained when we set the parameter K, denned in Eq. (16), equal to zero. In this limit, w and 7 also go to zero. Then, from Eqs. (18) and (19), we recover the GR force-free linear r-mode solution.

5. Discussion of the results and future directions In this work we have presented the analytical r-mode solution to the linearized Newtonian hydrodynamic equations with the GR reaction force. The velocity perturbations 6^v are proportional to e t ( 2 * + w ° t ), w j t n Wo = —40/3. Thus, they have the same sinusoidal behavior and the same frequency wo as the GR force-free velocity perturbations given by Eq. (11). The amplitude of the velocity perturbations is proportional to exp{—mi). Since w < 0, the GR force induces then an exponential growth in the rmode amplitude. The e-folding growth timescale TQR = 1/ro agrees with the GR timescale (1) found in Refs. [2,3]. The velocity perturbations 6^v contain also a piece proportional to a^y(A — l)fi 6 , where A is an arbitrary constant fixed by the choice of initial data. If we choose this constant A to be of order unity, then this part of the solution could be neglected 4 . A quite interesting feature that has emerged from recent investigations on r-modes is the presence of differential rotation induced by the r-mode oscillation in a background star that is initially uniformly rotating. That differential rotation drifts of kinematical nature could be induced by r-mode oscillations of the stellar fluid was first suggested in Ref. [7]. The existence of these drifts was confirmed in numerical simulations of nonlinear r-modes carried out both in general relativistic hydrodynamics 8 and in Newtonian hydrodynamics 9 . Differential rotation was also reported in a model of a thin spherical shell of a rotating incompressible fluid10. Recently, an analytical solution, representing differential rotation of r-modes that produce large scale drifts of fluid elements along stellar latitudes, was found within the nonlinear Newtonian theory up to second order in the mode amplitude and in the absence of GR reaction 11 . This differential rotation plays a relevant role in the nonlinear evolution of the r-mode instability 12 . Two questions could be then naturally raised, namely, is differential rotation also induced by the gravitational radiation reaction and does this differential rotation

174 play a relevant role in the nonlinear evolution of the r-mode instability? One of the aims of the investigation carried out in Ref. [4] is to initiate a programme that hopefully will allow to answer this question. The natural continuation of this investigation is then to try to find an analytical r-mode solution of the nonlinear hydrodynamic equations with the GR reaction force. This work is in progress 13 . Acknowledgments It is a pleasure to thank Kostas Kokkotas and Luciano Rezzolla for interesting discussions. This work was supported in part by the Fundagdo para a Ciencia e a Tecnologia (FCT), Portugal. OJCD acknowledges financial support from FCT through grant SFRH/BPD/2004. References 1. N. Andersson, Astrophys. J. 502, 708 (1998). 2. L. Lindblom, B. J. Owen and S. M. Morsink, Phys. Rev. Lett. 80, 4843 (1998). 3. N. Andersson, K. D. Kokkotas and B. F. Schutz, Astrophys. J. 510, 846 (1999). 4. O. J. C. Dias and P. M. Sa, Phys. Rev. D 72, 024020 (2005). 5. L. Blanchet, Phys. Rev. D 55, 714 (1997). 6. L. Rezzolla, M. Shibata, H. Asada, T. W. Baumgarte and S. L. Shapiro, Astrophys. J. 525, 935 (1999). 7. L. Rezzolla, F. K. Lamb and S. L. Shapiro, Astrophys. J. Lett. 531, L141 (2000). 8. N. Stergioulas and J. A. Font, Phys. Rev. Lett. 86, 1148 (2001). 9. L. Lindblom, J. E. Tohline and M. Vallisneri, Phys. Rev. Lett. 86, 1152 (2001). 10. Yu. Levin and G. Ushomirsky, Mon. Not. R. Astron. Soc. 322, 515 (2001). 11. P. M. Sa, Phys. Rev. D 69, 084001 (2004). 12. P. M. Sa and B. Tome, Phys. Rev. D 71, 044007 (2005). 13. O. J. C. Dias and P. M. Sa, in preparation (2005).

Space Radiation: Effects and Monitoring

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PARTICLES FROM T H E S U N

D. MAIA CICGE, Faculdade de Ciencias da Universidade do Porto, Observatorio Astron'omico, Alameda do Monte da Virgem 4430-146 Vila Nova de Gaia, Portugal E-mail: [email protected]

Particle acceleration is ubiquitous in all forms of coronal activity. The question is where, when, and how are those particles accelerated? In the Sun this question can be answered by using spatially resolved observations of the electromagnetic emissions and in-situ measures of plasmas in the heliosphere. In order to relate the remote and the in-situ observations one needs to consider both the problem of the origin of the particles, and of the propagation of those particles. We present here the details of a kinematic approach to the particle propagation problem that allows us to infer the particle injection profile at the Sun from in-situ measures at any point in the heliosphere.

1. Introduction Flares are the best observed energy input in the solar corona. The energy involved in a large flare is of the order of 10 32 erg. A large fraction of that energy resides in the accelerated electrons and ions. In solar active regions processes not yet identified accelerate typically electrons to a few hundreds of keV and protons to tens of MeV, although some events show signatures of much higher energies. Accelerated particles interact with the coronal plasma and emit radiation in the a broad wavelength range that includes not only gamma and X-rays, but also microwaves and radio. With a similar energy budget but very different in nature are Coronal mass ejections (CMEs). CMEs consist on a large volume of coronal magnetic fields and plasma (about 10 15 g) that leaves the Sun with a velocity that can exceed 2000 k m s - 1 . While in flares the energy is "spent" in heating the plasma and accelerating particles, in CMEs the energy is essentially in the bulk plasma motions. CMEs are relevant for particle acceleration because they drive shocks that can accelerate electrons and protons up to a certain energy. At 1AU that energy is about 1 MeV/nucleon, few keV/electron, close to the Sun is probably higher but it is not known.

177

178

IUJ

I I I V | I I I I I I I I I I | - p

I I I I I I I

HO4 1500

I' I' I' I ' I' I' I' I '

v,»

Ihooo >

" \

s

: io3

> 6

^500

I

•'

-1.0 -0.5 0.0 0.5 1.0 cosine of pitch angle

102

14:00

15:00

16:00

17:00 UT (2001 April 15)

18:00

•

19:00

20:00

Figure 1. One-minute integration 178-290 KeV electron flux as seen by the EPAM experiment on the ACE spacecraft. The lower-left-corner box shows the flux as a function of cosine of pitch-angle for one minute integrations at times before the event onset (filled circle) and shortly after the onset (open circles).

Particles accelerated close to the Sun can gain access to open magnetic field lines and be detected by telescopes on spacecraft, originating the so called in situ particle events. Comparing in situ measures of charged particles with observation of electromagnetic emissions at the Sun is not straightforward because particles are affected by a series of propagation effects as they move throughout the heliosphere. We will be describe in some detail a kinetic treatment of particle propagation that allows one to determine, from in situ particle measurements, the particle injection function at the Sun. That injection function can then be used to establish which phenomena at the Sun is more likely to be at the origin of those particles.

2. Modelling propagation effects One propagation effect to consider is the fact that the Sun rotates and as such magnetic field lines have a spiral nature. The length of that spiral depends on the Solar wind velocity: in the Earth vicinity it varies from about 1.1 to 1.4 AU. The magnetic field strength decreases with distance and conservation of the first adiabatic invariant will focuss the particles along the magnetic field direction as they propagate outwards. Small-scale irregularities in the magnetic field will the cause particles pitch angle to scatter. To determine the role of those effects on the particle profiles de-

179

10000

i|

T

I l I l IJ

JK•a

100

CO 13 CT

•o CD O

10

J

01

v

I

0.1 mean free path (AU)

13:40 13:50 14:00 14:10 14:20 UT (April 15, 2001)

Figure 2. (left) Quadratic deviations in the anisotropy (model minus observations) versus mean free path. Note the pronounced minimum at about 0.04 AU. (right) The injection function corresponding to the best-fit mean free path. See text for details.

tected by particle telescopes onboard spacecraft we use a kinetic approach similar to that described in Ref. 2. We generate randomly a few million test particles, that are "released" simultaneously from near the Sun surface in a spiral magnetic field and "followed" for a few days, considering the effects of adiabatic focussing and pitch angle scattering. Different models are characterized by distinct values of the mean free path A, and for each we compute the flux, and also the total anisotropy in pitch angle distribution, as a function of time and position. To illustrate the results of the modelization we will infer the injection function for EPAM 1 electron observations on 2001 April 15. Figure 1 shows the electron flux measured by the EPAM instrument in the 178-290 KeV energy range. The 2001 April 15 event is clearly anisotropic, but the PADs in the few minutes following the rise are not comparable to the instrument resolution: the flux is seen rising at all pitch angles, including the channels in the anti-solar direction. This is and indication of considerable scatter in the immediate vicinity of the Earth. In order to determine the mean free path A and injection function / we used a two step procedure. First, for each value of A we get the injection function /(A) that reproduces the observed flux. No assumption is made on the injection function shape or duration. Then, for each pair A and /(A) we compute the predicted average pitch and agle anisotropy and quantify its deviations from the observed anisotropy. This fitting procedure was fully tested for a wide range of artificial injection functions to be sure that is robust, efficient and unbiased. The results of this

180 fitting procedure for the 2001 April 15 event are shown in Fig. 2. As we can see the average quadratic deviations have a rather pronounced minimum for a mean free path between 0.04 and 0.05 AU. The injection function corresponding to these rather small mean free paths is of relatively short duration, less than 10 mn and is shown in Fig. 2. The fluxes of ~ l G e V protons seen by ground based neutron monitors on 2001 April 15 were modelled by Ref. 3, taking in account propagation effects, including scatter. They find a release time of 13:42±lmn UT and obtain an injection profile at the Sun that agrees with the electron injection function obtained for EPAM data. The onset of this particle event was also determined by Ref. 4 by comparing the arrival times of ions and electrons of different energies, the so called 1/v method. The ions and electrons onset times fall roughly on the same line, with an estimated release time at the Sun of 13 : 44 ± lmn UT, in remarkable agreement with ours and Ref. 3 model results. 3. Conclusion The particles detected in situ by experiments onboard spacecraft are affected by propagation effects. We have accounted for those effects and developed a procedure to recover not only the release time but also the duration and shape of the injection function. The 2001 April 15 event was selected to illustrate the method because its characteristics imply a rather small mean free path at electron energies of a few hundred keV. We have obtained a good agreement with relase times obtained from our model with results from other authors, using other methods, namely the arrival of particles of different energies 4 . Acknowledgements D. Maia was partially supported by Fundagao para a Ciencia e Tecnologia, through the program POCTI/FNU/43776/2001 and through grant SFRH/BPD/5521/2001. References 1. 2. 3. 4.

R. Gould et al, Sol. Phys. C34, 1729 (1995). R. Vainio, Kocharov Astronomy and Astrophys. 25, L527 (2000). V. Bieber, Astrophys. J. A632, 287 (2003). A. Tylka, Phys. Rev. Lett. 86, 4492 (2003).

SIMULATIONS OF SPACE RADIATION M O N I T O R S

B. T O M E * LIP, Av. Elias Garcia, 14-1 1000 Lisboa, Portugal E-mail: [email protected]

The radiation monitors to be included in the payloads of future space missions, must satisfy the scientific and safety requirements while meeting mass and power severe constrains. GEANT4 is a powerful tool to perform the development and the optimization of such detectors thanks to its capabilities for describing complex detector geometries and simulating the passage of particles through matter. The inclusion of an optical physics process category allows the full simulation of scintillation based detectors. A GEANT4 based simulation of a compact lightweight radiation monitor concept for future space missions is presented.

1. Introduction Radiation monitors are becoming an essential component in space missions, providing crucial radiation environment information for the in-flight protection of the spacecraft and instruments onboard. In addition, the data acquired during the mission are a valuable input for space environment models; in particular data gathered in missions to other planets (e.g. particle fluxes and spectral distributions as a function of the distance to the Sun) are essential for models describing the injection and propagation of particles from the Sun. Several of the future space missions (e.g. LISA, Gaia, JWST, BepiColombo) are planned to carry radiation monitors. Given the limited resources on mass, power and accommodation on-board of a spacecraft, a new generation of compact and lightweight general purpose energetic particle detectors are being explored. Monte Carlo simulation tools are essential to develop and optimize a given detector concept and to explore alternative detector concepts. The ability to predict the detector performance in realistic radiation environments is also crucial. This requires the implementation of software models •This work was supported through F C T grant SFRH/BPD/11547/2002.

181

182

allowing to perform an end-to-end simulation, including the particle propagation from the source to the spacecraft location, followed by a detailed simulation of the detector response. The GEANT4 toolkit is an ideal framework for simulating the particle interactions with the detector materials in complex geometries. A GEANT4 based simulation of a compact lightweight radiation monitor concept for future space missions is presented.

2. Space radiation environment The high energy ionizing radiation 3 environment in the solar system consists of three main sources: the radiation belts, galactic cosmic rays and solar energetic particles. The radiation belts are composed of electrons and protons trapped in the magnetosphere of the planets. In the radiation belts of the Earth (Van Allen belts) electrons have energies smaller than about 6 MeV and protons have energies up to 250 MeV (see Ref. 1). The Earth radiation belts are mostly relevant for low orbit missions. The galactic cosmic rays (GCR) component consists of a continuous flux of electrons, protons and ions, arriving from beyond the solar system, which are originated mainly in the supernovae. The GCR flux is maximum at energies around 1 GeV/n, decreasing as a power law for higher energies. The lower energy particles are strongly affected by the 11-year solar cycle. Although the GCR flux is comparatively low, energetic heavy ions can locally produce significant energy depositions originating single-event upsets. Solar energetic particles (SEP) events consist of a sudden and dramatic increase in the flux of particles from the sun, including electrons, protons and heavier ions. SEP events are associated with impulsive solar flares and Coronal Mass Ejections and occur with higher probability during the maximum of the solar cycle. Although the energies of SEP rarely exceed few hundred MeV they give rise to potential risks for space missions due to the high fluxes attained. On the other hand their study is relevant for developing and testing solar particle propagation models (Ref. 2).

a

I n the context of space radiation effects, particles are considered energetic if their energy is high enough to penetrate the outer skin of the spacecraft. This means electrons with energies above 100 keV and protons and ions with energies above 1 MeV.

183 3. The G E A N T 4 simulation toolkit GEANT4 (Refs. 3, 4) is a Monte Carlo radiation transport simulation toolkit, with applications in areas as high energy physics, nuclear physics, astrophysics or medical physics research. It follows an Object-Oriented design which allows for the development of flexible simulation applications. GEANT4 includes an extensive set of electromagnetic, hadronic and optics physics processes and tracking capabilities in 3D geometries of arbitrary complexity. The electromagnetic physics category cover the energy range from 250 eV to 10 TeV (up to 1000 PeV for muons) while hadronic physics models span over 15 orders of magnitude in energy, starting from neutron thermal energies. The optical physics process category allows the simulation of scintillation, Cherenkov or transition radiation based detectors. A distinct class of particles, the optical photons, is associated to this process category. The tracking of optical photons includes refraction and reflection at medium boundaries, Rayleigh scattering and bulk absorption. The optical properties of a medium, such as refractive index, absorption length, reflectivity coefficients, can be expressed as a function of the photon's wavelength. The characteristics of the interfaces between different media can be defined using the UNIFIED optical model (Ref. 5). Full characterization of scintillators include the emission spectra, light yield, the fast and slow scintillation components and associated decay time constants. 4. Simulation of a generic space radiation concept The simulation of a simple space radiation monitor concept (Ref. 6) was implemented using the GEANT4 toolkit. The detector consists of a tracker made of two position sensitive silicon planes followed by a scintillating crystal surrounded by photodetectors (Fig. 1), An aluminum foil placed on the backside of the crystal is used to tune the energy range of the detector. In the present simulation the silicon tracker planes were 0.065 mm and 0.5 mm thick, the crystal size was 3 x 3 x 3 cm 3 and the aluminum absorber was 2 mm thick. The scintillation properties of the CsI(Tl) crystal were used. These included a light yield of 65000 photons per MeV of deposited energy and two scintillation components with decay time constants of 0.68 /is (64%) and 3.34 /is (36%). The scintillation light is readout by large area silicon PIN photodiodes, with a spectral sensitivity matching the CsI(Tl) emission spectrum. Since the silicon photodiodes are sensitive both to photons and charged particles, this makes them suitable to be used also as an anti-coincidence shield.

184

Figure 1. View of the simulated particle detector. From left to right are shown the two silicon planes, the scintillating crystal surrounded by photodetectors and the aluminum plane followed by a charged particle sensitive detector.

In the left plot of Fig. 2 is shown the fraction of the initial particle energy deposited in the crystal for protons of energies between 25 MeV and 125 MeV. Protons with energies of 25 MeV or 50 MeV are absorbed in the crystal. For 100 MeV protons the effect of straggling is visible.

•'tMtfilM'

Figure 2. Left - Total energy deposited in the crystal by incident protons, expressed as the fraction of the initial particle energy. Right - Simulated time spectrum of the hits in the photodetectors.

185 The principle of the anti-coincidence operation mode is illustrated in the rigth plot of Fig. 2. It shows the time of the simulated hits in the photodiodes, produced by a 100 MeV proton that is not stopped inside the crystal. The fast signal at time zero is due to the direct ionization produced by the proton in a photodetector, while the slower signal comes from the scintillation light emitted by the crystal. Thus by a suitable signal processing it is possible to disentangle the direct ionization from the scintillation. This allows to identify the events where particles entered the detector from the sides or were not fully absorbed in the crystal. Particle identification is performed by measuring the energy loss in the thin silicon trackers. Figure 3 shows the simulated energy loss in the first silicon plane, as a function of the initial particle energy, for electrons, protons and alphas.

Incident Energy (MeV) Figure 3. Energy deposited in the first silicon layer by electrons, protons and alphas, as a function of the initial particle energy.

References 1. E.R. Benton and E.V. Benton, "Space Radiation dosimetry in low-Earth orbit and beyond", Nucl. Instrum. Methods B184, 255 (2001). 2. See talk by D. Maia, these proceedings. 3. S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce et al., "GEANT4 - a simulation toolkit," Nucl. Instrum. Methods, A 506, 250 (2003). 4. M. G. Pia, "The Geant4 Toolkit: simulation capabilities and application results," Nucl. Phys. B - Proc. Suppl, 125, 60 (2003). 5. A. Levin and C. Moisan, "A more physical approach to model the surface treatment of scintillation counters and its implementation in DETECT," in Proc. IEEE 1996 NSS, 2 (1996). 6. A. Owens et al. (ESA/ESTEC), private communication.

GEANT4 DETECTOR SIMULATIONS: RADIATION INTERACTION SIMULATIONS FOR THE HIGHENERGY ASTROPHYSICS EXPERIMENTS EUSO AND AMS* PATRICIA GONCALVES LIP-Laboratorio de Instrumentacdo e Fisica Experimental de Particulas, Av.Elias Garcia n°14 - 1", 1000-149 Lisboa The system architecture of a GEANT4 based simulation framework and its application to EUSO/ULTRA and AMS/RICH performance studies are presented. ULTRA (Ultraviolet Light Transmission and Reflection in the Atmosphere) is an experimental support activity of EUSO (Extreme Universe Space Observatory), an experiment devoted to the study of extreme energy cosmic rays and neutrinos. Relevant aspects of the ULTRA simulation, namely the description of optical processes and the simulation of Fresnel lenses using parameterisation/replication techniques are described. The RICH (Ring Imaging CHerenkov detector) of the AMS (Alpha Magnetic Spectrometer) experiment, will incorporate a dual radiator, made of a low refractive index material, aerogel, and of sodium fluoride (NaF). A more realistic description of Cherenkov photon transmission through the aerogel surface, based on Atomic Force Microscopy images, was implemented in GEANT4.

1. The simulation framework GEANT4 [1,2] is a toolkit for the simulation of particle transport and interaction with matter, featuring namely: the description of geometries of arbitrary complexity, standard electromagnetic physics processes (photons, electrons, positrons and muons). It describes Optical physics processes including the generation of photons by Scintillation, Cherenkov and transition radiation effects, Rayleigh scattering and media-boundary interactions (reflection, refraction). GEANT4 is based on an Object Oriented design, allowing the implementation of flexible simulation applications and of new/upgraded physics processes. The GEANT4 based part of the simulation framework entitled "GEANT4SpaceApplication" consisted of the description of different AMS and EUSO related detector geometries, of the interface with alternative sets of primary event generators and of the radiation transport processes. As for the integration of the readout electronics, signal digitisation and event reconstruction, it was done via the DIGITsim module [3] .The object oriented technology * This work was supported by FCT under Grant SFRH/BPD/11500/2002.

186

187 for event data persistency and data analysis was handled through ROOT, LCG PI/AIDA and POOL. 2. The ULTRA experiment simulation Within the EUSO experiment [4] there is an on-going program of critical design parameter studies, implying a set of dedicated UVscope experiments. The detection of the Cherenkov light associated with the Extensive Air Showers (EAS), measuring UV light diffusion coefficients of different types of media at the surface of the Earth, is the main goal of the ULTRA project (ULTRA - UV Light Transmission and Reflection in the Atmosphere) [5]. The ULTRA detector is a hybrid Figure 1. Typical configuration of the system consisting of a UV detection system, the ULTRA experiment. The area UVscope, sensitive to the diffused Cherenkov covered by the ETscope array is seen light from EAS, in coincidence an array of in the UVscope field of view. The UVscope is located on top of a hill scintillation detectors, the ETscope. A typical (represented by the thick line) at a configuration of the ULTRA experiment is height H. shown in Fig. 1. 2.1. Simulation of the ULTRA ground detectors Each ULTRA ETscope station consists of a NUCLEAR NE 102A plastic scintillator, with dimensions 80x80cm2 and 4cm thick, housed in a pyramidal stainless steel box. The inner walls of the box are coated with a white diffusing paint (similar to Oxide Titanium paint). The scintillation light is collected by two photomultipliers at approximately 30cm from the scintillator surface. The optical properties of the interface defined by the air and the white reflector covering the aluminum were specified within the UNIFIED model the "dielectricdielectric" TYPE and the "groundfrontpainted" FINISH were chosen. The scintillator emission spectrum, peaking at 423nm, and its light yield, about 10000 photons/MeV of deposited energy, were included in the material optical properties.

Figure 2. Schematic drawing of an ULTRA ETscope station.

188 2.2. Simulation of the ULTRA UV telescope The Uvscope consists mainly of a Fresnel lens and one photomultiplier (PMT) located in the focal plane, enclosed in a cylindrical aluminum housing. The lens is 457 mm in diameter and is made of UV transmitting acrylic with 5.6 grooves per mm. A wide band filter (300 to 400 nm) is used to reduce the background. The external housing consists of a 1mm thick aluminum cylinder, with a length of 1030mm and a diameter of 518mm. To describe the geometry of the lens a parameterised replication of G4Cons volumes used. Each lens groove is described as a fnistrum of a cone, with cross-section represented by a straight line with slope varying as a function of the radius. For this purpose the GEANT4 classes SpaceGEANT4FresnelLensParameterisation (inheriting from the G4VPVParameterisation class) and SpaceGEANT4FresnelLens were designed. A PMT with a 68mm diameter window was placed at a distance of 441.97mm from the lens, corresponding to the effective focal length at X=400nm. The UNIFIED model was chosen to describe optical processes at material interfaces. For the "G40pticalSurface" attributes of the Air/Aluminum interfaces, the "dielectricdielectric" TYPE and "groundfrontpainted" FINISH were used. All aluminum parts (cylinder walls and lens support) are assumed to have 5% reflectivity. Figure 3 represents the behavior of the UVscope when hit by a beam Ultraviolet photons.

Figure 3. Left: ultraviolet photons focused by the Fresnel lens into the PMT. Right: the photons out of focus cross the lens surface in more than one point.

3. Simulating the AMS RICH radiator The AMS (Alpha Magnetic Spectrometer) experiment aims at characterising cosmic rays before reaching Earth's atmosphere. Its main objectives are the search for anti-matter and dark matter and the study of the propagation and confinement of cosmic rays in the galaxy [6]. AMS had a precursor flight in June 1998 aboard the Discovery Space Shuttle, with around 100 million events recorded. The capabilities of the AMS spectrometer will be improved and extended through the inclusion both of new detectors and of a stronger magnetic

189 field. Lip's collaboration in AMS is centred in the RICH (Ring Imaging CHerenkov) detector. 3.1. RICH radiator sim ulation The AMS RICH will be built with a dual radiator, made of a low refractive index material, aerogel (n=1.03), and of sodium fluoride (NaF, n=1.33). It will provide an independent measurement of the velocity and . charge of the cosmic rays. The RICH is a complex detector and performance assessment depends critically on the correct modeling of the light production, transmission and collection. A simplified design of the AMS RICH radiator setup, consisting of aerogel tiles supported by a Figure 4. Implemented setup for plexiglas foil, was implemented in the *e aerogel radiator GEANT4 based simulation framework. A variable number of 11.3 x 11.3 x 11.3cm aerogel tiles, separated by a 0.1cm gap, were placed in a vacuum tank, of corresponding variable dimension, on top of a 0.1cm thick plexiglas foil. The gaps between the aerogel tiles can be alternatively left in vacuum, filled with plexiglas or with a material opaque to the Cherenkov photons. The implemented aerogel (Si02+vacuum) and plexiglas (C5H802) j " . ' ''| properties, refractive index, absorption length, f and clarity, in the case of aerogel, correspond to •'; the AMS RICH radiator specifications. The setup implemented is shown in for a 3x3 aerogel tile Rgure 5 cherenkov light array. This setup is being used to study more produced by one 80GeV realistic models describing photon scattering in electron crossing the RICH the aerogel surface, and to compare them with the results of the test beam of the AMS RICH prototype. In Fig. 4, shows the Cherenkov light produced by an 80GeV electron crossing the simulated radiator setup were also the effects of Rayleigh scattering and absorption can be observed. 3.2. AFM: Atomic Force Microscopy Previous test beam results revealed a disagreement between aerogel photon scattering in data and its description in the simulation. This effect, which cannot

190 be described by the UNIFIED model in GEANT4, is thought to come from the interference of the photons with the aerogel surface defects. Although the UNIFIED model allows for a rather detailed _ ,.«, description of photon reflection, this is not the case for transmitted photons, unfortunately the • •_, case of Cherenkov photons. Therefore, a more « "• " \ realistic description of the directional behavior \ ,' , '• ' ' >.» of the transmitted Cherenkov photons was .'$.'*'' implemented with basis on Atomic Force '. " •" . , Microscopy (AFM) scanning of the aerogel surface. In Fig. 6, a 2D view of the surface of i an aerogel sample is shown, where surface defects can be observed. Two approaches were F i g u r e 6 A t o m i c F o r c e Micros _ followed to obtain a realistic description within copy image of the surface of an aero el sam le GEANT4 of the direction of Cherenkov photons S P after crossing this type of surface. In one approach, a new virtual class "G4VUserMapBoundaryModel" interfaces the surface maps to GEANT4. The other approach consisted of extending the UNIFIED model parameterisation by providing new methods to the GEANT4 class "G40pBoundaryProcess". References 1. S. Agostinelli, J. Allison, K. Amako, J. Apostolakis, H. Araujo, P. Arce et al., "GEANT4 - a simulation toolkit," Nucl. Instrum. Methods, Vol. A506, 250 (2003). 2. M. G. Pia, 'The Geant4 Toolkit: simulation capabilities and application results," Nucl. Phys. B-Proc. SuppL, Vol. 125, 60 (2003). 3. M. C. Espfrito-Santo, P., Goncalves, M. Pimenta, P. Rodrigues, B. Tome and A. Trindade, "GEANT4 Applications for Astroparticle Experiments", IEEE Transactions on Nuclear Science, Vol. 51, 1373 (2004). 4. L.Scarsi, 'The Extreme Universe of Cosmic Rays: Observations from Space", II Nuovo Cimento, Vol. C 24, 471 (2001). 5. O. Catalano, P. Vallania, D. Lebrun, P. Stassi, M. Pimenta, and C. Espirito Santo, "ULTRA technical report," Tech. Rep. EUSO-SEA-REP001, 2001. 6. S. P. Ahlen, V. M. Balebanov, R. Battiston, U. Becker, J. Burger, M. Capell et al, "An antimatter spectrometer in space," Nucl. Instrum. Methods, Vol. A 350,351(1994).

SOFTWARE FOR RADIOLOGICAL RISK A S S E S S M E N T IN SPACE MISSIONS

A. T R I N D A D E , P. R O D R I G U E S LIP, Laboratorio

de Instrumentacao e Fisica Experimental Av. Elias Garcia 14-1, 1000-149 Lisboa, Portugal E-mail: [email protected]

de

Particulas,

The radiation environment found in space missions present a potential risk to astronauts as well to electro-mechanics components of spacecrafts. In this paper we will present an overview of main sources and composition of the space radiation environment with relevance to astronauts dosimetry. Types and effects of space radiation in astronauts and radiation protection limits are discussed. A discussion on software tools for dose prediction is presented, with focus on how medical applications can help to improve space radiation risk assessment for mission planning.

1. Introduction The radiation environment in space is characterized by a wide variety of primary particles covering an extended range of energies1. The three main sources of primary ionizing radiation are the galactic cosmic rays (GCR), trapped radiation in Earth's magnetosphere and solar particles events (SPE). GCR are mainly composed of protons (87%) and 7-rays, with a small fraction of electrons/positrons (2%) and heavy ions (1%). Trapped radiation mainly consists on electrons with energies up to 10 MeV and protons up to 100 MeV. Its presence is continous in time, with fluxes with variable intensity, function of Sun's solar activity. SPE radiation is composed of mostly low-energy protons and electrons, with fer percentage of higher-energy protons and heavy ions. In contrast to GCR, SPE are event driven, with occasional high fluxes over short periods, which coincide with increases in Sun's solar activity. The composition and intensity of these radiation sources can be modified by local planetary environments, distance of the location of exposure to the sun, existence of moons or local geology. Often, secondary ionizing radiation can be present, due to the interaction of the primary radiation with shielding materials (like the

191

192

spacecraft hull). Both primary and secondary particles can cause radiation damage in electronic components or biological damage when passing the body of an astronaut. The magnitude of the effects is however largely dependent on the type of space mission. GCR and SPE radiation are the main cause of concern for long-term mission duration, due to its constant presence (GCR) or high-Muxes (SPE). Severity of such event is largely enhanced by the fact that in long-term missions, immediate medical care would not be available. In order to provide an addition protection, shielded habitats have to be designed. Short-term missions in the Low-Earth Orbit are also associated with some risks, mainly caused by the trapped protons and electrons. Although the risk is reduced due to the shielding provided by spacecraft and habitat modules, extra-vehicular activities (EVA) present some concern due to the low attenuation provided by EVA spacesuits, against trapped electrons and protons.

2. Radiation Effects in Astronauts Galactic cosmic heavy ions encountered in space are the more dangerous to human health due to their high linear energy transfer, which can break both strands of DNA. Besides this direct action against the cell DNA, an indirect action also takes place. This happens when after the irradiation, water molecules become ionized and form highly reactive free radicals. The free radicals can interact with strands of DNA causing them to break. In some conditions, mechanisms of DNA self-repair fail, leading to the appearance of cell mutations or cell death. Depending on the exposure and type of the incident radiation, the effects on human health are classified in short and long term. Short term effects range from nausea and damage to the central nervous system and death (symptoms manifest within minutes to 30-60 days. Long term effects (symptoms do not manifest for months or even years) include the development of cataracts and cancer. In order to correlate radiation exposure with molecular, biochemical and somatic effects several physical parameters were defined: absorbed dose, dose equivalent and effective dose. Absorbed dose is defined as absorbed energy per unit mass. In order to account for the different biological effectiveness of the incident radiation, the unit dose equivalent is used. Dose equivalent for a given tissue is given by the absorbed dose weighted by a quality factor Q. The normalization factor can range from 1 to X-rays, 7 -

193 rays, /3-rays to 20 for high-energy protons, a-particles, fission fragments and heavy nuclei. In order to account that for the same radiation type, different tissues and organs have their one radio-sensitivity value, the unit effective dose was denned. It is obtained by multiplying the individual dose equivalents by risk weighting factors. These factors reflect different organs radio-sensitivity to developing cancer and they represent the dose that the total body could uniformly receive that would give the same cancer risk as various organs getting different doses. In Table 1 the maximum dose equivalent admissible for U.S.A astronauts in short-term missions are shown. Data taken from Ref. 2. Table 1. Ionising radiation exposure short-term limits (for U.S.A. crewmembers). Bone marrow (Gy-Eq.)

Eye (Gy-Eq.)

Skin (Gy-Eq)

Annual

50

2.0

3.0

30 days

0.25

1.0

1.0

Table 2 presents the effective dose limits for U.S.A astronauts. These limits provide the maximum effective dose that an astronaut with a given age at the time of exposure can receive without increase in more than three percent the risk of developing radiation-induced disease. Data taken from Ref. 2. Table 2. U.S.A career effective dose (in Sievert) limits (based on three percent excess lifetime risk) Age at exposure

Female (Sv)

Male (Sv)

25

0.4

2.0

35

0.6

1.0

45

0.9

1.5

55

1.7

3.0

3. From Medical to Space Applications In order to guarantee that the current exposure limits are respected, dose predictions are required for mission planning. Such previsions need to taken into account effects such as the modulation effects in space-radiation envi-

194

ronment (e.g. solar cycle), spacecraft shielding, spacecraft orbit and location. Besides space dosimetry, the need to predict dose is a common requirement of several research fields such as external and internal radiotherapy, nuclear medicine imaging, radionuclide therapy and hadron therapy. Knowledge off the limitations presented by semi-analytical algorithms have favored the introduction of Monte Carlo techniques in dose calculations, since they allow a detailed treatment of the physics processes involved in the interaction of ionizing radiation with matter 3 . Geant4 is an object-oriented Monte Carlo simulation toolkit 4 , developed by the RD44 (up to 1998) and Geant4 Collaboration, which provides a new approach for development of accurate dose calculations applications. Geometry modeling capability is also of paramount importance for realistic and efficient dose calculations, as required for astronauts space dosimetry. In this area, Geant4 provides a large set of solids of different complexity and allows an efficient repetitive structure representation. Such feature is vital for the implementation of realistic voxel-based anthropomorphic geometries which are used to define the setups for particle transport. In these phantoms, the human body is modeled as a large set of small elements with different chemical and density properties. Several applications, using this Geant4 feature were developed and they show how an important requirement of dose calculations in astronauts can be fulfilled. Example applications include the development of a Geant4-based framework for dose calculation in radiotherapy with external beams 5 and the Clear-PEM simulation framework for realistic scenarios for a Positron Emission Mammography detector optimization 6 .

3.1. Cleat—PEM Positron

Emission

Mammography

System

The Clear-PEM detector is a positron emission mammography planar system designed to image both the breast and axillary lymph nodes. In order to optimize the final detector layout, realistic estimations of detector parameters, like detector efficiency, spatial resolution and background rate, must be obtained. A modular Geant4-based framework was developed, which includes the simulation of radioactive decay and photon tracking in defined phantoms, detector response, signal formation in the associated readout front-end electronics, and data acquisition by the DAQ/trigger system. Currently implemented phantoms in this module are a mathematical type torso-phantom, defined by simple mathematical volumes, and a voxelized NCAT phantom (developed by W.P Segars7) which models in detail

195 a human body. The level of detail than can be included in the simulation can be observed from the following Tenderized images - Fig 1. Lung \

^Ribs \

) J ' - Heart

V

Liver

-~

-

— Stomach

5

h Spleen /• / Kidney

X..

Spine 35.2 cm

26.7 cm ~* »Figure 1. Volume rendering of the implemented NCAT torso phantom in Geant4 with 3.25 x 3.25 x 3.25 m m 3 voxel resolution.

4. Conclusions Dose predicament in astronauts is a complex task that requires both the correct description of space environment, simulation of shielding material provided by spacecraft, habitat and EVA suits as well as high resolution human body models within the spacecraft. Several developed applications for Monte Carlo dose calculations in Medicine have shown that this requirements can be fulfilled, making possible to improve the precision of dose calculation in space. Acknowledgments The authors would like to thank colleagues from the ClearPEM collaboration for their suggestions and contributions. References 1. E.R. Benton and E.V. Benton, Nucl. lustrum. Methods Phys. B184 255 (2001). 2. Effects in Biological Material, ECSS ECSS-E-10-12 Draft 0.5. 3. P. Andreo, Phys. Med. Biol. 37 861 (1991). 4. S. Agostinelli et al, Nucl. lustrum. Methods Phys. A506 250 (2003).

196 5. P. Rodrigues, A. Trindade, L. Peralta, C. Alves, A. Chaves and M.C. Lopes, Appl. Rad. Isotope 61 1451 (2004). 6. P. Lecoq and J. Varela, Nucl. Instrum. Methods Phys. A486 1 (2002). 7. W.P. Segars, Development of a new dynamic NURBS-based cardiac-torso (NCAT) phantom, PhD dissertation, The University of North Carolina, USA 2000.

Neutrino Physics

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RESULTS FROM K2K

S. A N D R I N G A IFAE/UAB F O R T H E K2K COLLABORATION The K2K experiment was designed to confirm or disprove the atmospheric neutrino oscillations. Its experimental concept and detectors are described together with the full analysis of neutrino oscillations. The K2K data confirms neutrino oscillations, excluding the null oscillation hypothesis at 4
1. The K2K Experiment The KEK to Kamioka neutrino oscillation experiment, K2K, is the first neutrino long base line experiment. It was designed to confirm or disprove, in a controlled way, the neutrino oscillations of v^ to vT, first seen by SuperKamiokande 1 in athmospheric neutrino events. A 98% pure muon neutrino beam with a known energy spectrum, peaked at 1.3 GeV, is produced at the KEK laboratory and detected at a distance of 250 km at the Super-Kamiokande detector. The fixed travelled length (L) and the reconstruction of the neutrino energy (Ev) at both sites allows for a direct determination of the oscillation pattern of the v^ survival probability - which for two neutrinos oscillations is given by: Piy» -» v„) = 1 - sin2(26>) sin 2 (1.27Am 2 L/£„),

(1)

where Am 2 is the mass difference of the two neutrino mass eigenstates involved in the atmospheric neutrino oscillations and 6 is the mixing angle between the mass and the weak eigenstates. With 10 20 protons on target (POT) at the K2K beam line, similar contributions of statistical and systematic uncertainties are expected to be achieved in the oscillation parameters. Since Super-Kamiokande is used as the far detector and the K2K spectra is similar to that of atmospheric neutrinos, the large sample of atmospheric neutrino events can be used to control detection systematic uncertainties at the far detection site. On

199

200

the other hand, a set of detectors at the production site collects very high statistics to reduce the systematic uncertainties on neutrino production, namely the beam intensity and energy spectra, and on neutrino interactions. At the near site there are thus one water Cerenkov detector, to cancel detection uncertainties common to Super-Kamiokande; a set of Fine Grained detectors, to study the modelling of neutrino interactions and reduce the corresponding systematic uncertainties; and a set of beam monitors to verify, and control the neutrino spectra and flux and its extrapolation to the far site. A detailed description of the K2K experiment is given in Ref. 2, and a schematic view of the near detector site is shown in Fig. 1.

Figure 1. The Near Detector complex of K2K. From upstream (left) to downstream, there are the 1KT Cerenkov detector, the SciFi and SciBar fine grained detectors and the Muon Range Detector.

1.1. Gerenkov

detectors

The Super-Kamiokande (SK) 1 is a 50 kt water Cerenkov detector with 40% of its area covered by PMTs; the outer part of the detector is used for vetoing cosmic-ray muons, leaving a fiducial volume of 22.5 kt. The K2K flux intensity is 106 neutrinos per spill. The charged current interactions (CC) producing a muon are detected with a 93% efficiency while neutral current interactions (NC) are detected with an efficiency of 68%.

201 At the near site, there is a water Cerenkov with a total volume of 1 kt, the 1KT detector, with an angular light coverage identical to that of SK. The fiducial volume is of 25 t but the flux intensity is 10 11 neutrinos per spill. The efficiencies are of 87% (CC) and 55% (NC), lower than those of SK due to the presence of lower energy neutrinos (outside the SK acceptance) and an effective upper limit on the neutrino energy in CC events. This upper limit comes from the requirement that the event be totally contained in the fiducial volume, so that the muon momentum (PM) can be reconstructed - it corresponds to PM < 1.5 GeV/c (Eu <2 GeV). The same reconstruction and analysis methods are used for both Cerenkovs and the results at the 1KT give the prediction for the total number of events at SK. For energy reconstruction, events with only 1 Cerenkov ring are selected and this ring is required to be compatible with that produced by a muon. This results on a sample composed mostly of CC quasi-elastic (QE) events, for which Ev can be reconstructed as: _ (mN - V)Eil - pl/2. + mNV - V2 (mN-V)-E^+plicos9fi '

[

>

where TUN is the free neutron mass, V is the nuclear binding energy, and the Fermi motion of the neutron is neglected. For 40% of the events, which are not QE, the reconstructed energy is biased to lower values. 1.2. Fine grained

detectors

The fine grained detector complex is composed by two complementary scintillator trackers and a Muon Range Detector (MRD a3 ) to contain the escaping muons so that their momentum (from 0.55 to 3.5 GeV/c) can be measured. In SciFi4 the neutrinos interact predominantly in water, as at SK, while SciBar5 is a fully active detector for neutrino interactions in carbon - allowing for better reconstruction of the interaction vertex. The low threshold of the scintillators (for a proton, the momentum Pp >450-650 MeV/c) allows for the selection of large samples with a muon and a second track. Assuming QE kinematics, the direction of the second track can be compared with the one predicted from the measurement of the muon, A6P. The CC events are thus separated in three samples: 1 track, 2 track QE-like (A0 P <25°) and 2 track nonQE-like. a

D u e to its high density, the MR.D is also used to monitor the stability of beam intensity, spectra and direction.

202

,§1000 in

o 800 9 600 400 200

'0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 GeV r

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 GeV r 2 track non-QE •

0 Data MC CCQE

£fi^

0.2 0.4 0.6 0.8

1.2 1.4 1.6 1.8 2r GeV

Figure 2. Distribution of q2 (reconstructed assuming QE kinematics) in the 1 track, 2 track QE-like and 2 track nonQE-like samples selected by SciBar. The data (dots) are compared to the total expectation from the K2K simulation (open histograms). The QE fraction is shown by the shaded histograms.

203

Figure 2 shows the three samples selected in SciBar, with different QE purities. A clear disagreement is seen between the data and the expectations from simulation in the region of low transfered energy between the leptonic and hadronic currents (q2), and for the samples dominated by nonQE interactions. This disagreement is seen in all K2K sub-detectors. Its origin is under study but it was shown that it can be either due to an excess in the prediction of coherent neutrino interaction with the nuclei (which is extrapolated down from higher energies) or to a lack of nuclear corrections in the production of A resonances (which are harder to calculate theoretically than those for the QE interaction). 1.3.

Simulation

K2K uses two base simulations in its analyses: for the neutrino production at the K2K beam line and for the neutrino interaction in the K2K subdetector targets. The K2K beam is produced by sending 12 GeV protons from the KEK Proton Synchrotron into an Aluminium target; two magnetic horns followed by a decay volume focus the produced 7r+ which will decay to fi+ and v^. The fi+ are later absorbed while the v^ follow the direction of SK. This is simulated as described in Ref.6 and the main uncertainties come from the hadron production at the target. The neutrino interactions with the nuclei of the different detectors are simulated with the NEUT package7, also used by the Super Kamiokande collaboration. It includes the description of QE interactions 8 , resonance production 9 , deep inelastic scattering 10 and coherent interactions 11 . It uses the Fermi gas model to implement the nuclear corrections and includes final state interactions such as proton rescattering or pion absorption in the nuclei. 2. Neutrino Oscillation Analysis The distribution of P^ and 6^ in the 1-track sample of 1KT, and the three samples of SciFi and SciBar are all used together in a fit to determine corrections to the K2K simulations. The fit is done first without the region of low #M (i.e. the low q2 region, where there is the discrepancy between data and expectations), to fix the experimental parameters - corrections to the K2K energy spectra and absolute energy scales of the near detectors - and then, including these events, to determine the ratio of nonQE to QE cross-sections. This last step is done under two different assumptions

204

- the absence of coherent interactions or the suppression of A resonance production by g 2 /0.1 for q2 < 0.1 - which give similar quality fits. The corrected description of the simulated spectra and interaction models is used to extract the predictions at SK b . At SK the events are selected following the atmospheric analysis 1 . The pulsed structure of the K2K beam allows for a very precise time selection, reducing the atmospheric neutrino background to a rate of 2 per mil. Table 1. SK no oscillation corresponding 47.9xl0 1 8 and

Event Selection, data and expectations in the case of are shown separately for two K2K periods, the second to half light coverage of SK and addition of SciBar, with 41.2xl0 1 8 P O T , respectively. DATA (I+II)

MC no osc. (I+II)

FC 22.5 kt

107 (55+52)

150.9 (79.1+71.8)

for Normalization

multi-ring

40 (22+18)

56.9 (30.2+26.7)

NC enhanced

1-ring

67 (33+34)

94.0 (48.9+45.1)

e-like

10 ( 2+ 7)

8.6 ( 4.0+ 4.5)

NC 7r° dominated

fi-Wke

57 (30+27)

85.4 (44.9+40.5)

for Spectra Shape

CC enhanced

The number of events seen and predicted in the absence of oscillations are shown in table 1, giving a clear indication of v^ disappearance. The compatibility with the null-oscillation hypothesis is 0.28% from the total number of events (107 events seen with 150.9 expected) and 0.11% considering only the shape of the energy spectra reconstructed in the sample of 1/x-ring events (dominated by QE), shown in Fig. 3. 2.1. l/fj, —> vT

oscillation

The determination of the oscillation parameters is done by a likelihood fit using the two informations simultaneously. The likelihood includes three terms L = Lnormaiization(sm2 (20), Am2, 2

2

LE„ shape (sin (29), Am , Lsysternatics (systematics). b

systematics).

systematics). (3)

T h e suppression of A resonance production is used in what follows, but the oscillation results do not change if the suppression of coherent interactions is used instead.

205

> d c

>

Figure 3. E„ reconstructed according to Eq. 1, for Data (dots) are compared with the shapes expected (dashed line) and the hypothesis of sin 2 (20) = 1.00 line). In both cases the expectations are normalized

the 1^-ring sample selected at SK. in the hypothesis of null oscillation and Am2 = 2.79 x 1 0 - 3 eV 2 (full to the number of events in data.

There are no constraints in the values of the neutrino oscillation parameters, while the systematic uncertainties include 5% from the extrapolation of the K2K flux and spectra to the far site, 5% from the overall normalization dominated by the fiducial volume determination in the Cerenkovs; 0.5% from the near detector flux and spectra determination and 0.5% from the nonQE to QE interaction cross-sections; and the experimental uncertainties on SK efficiency and energy scale, both of around 3%. The best fit gives compatible numbers for the systematic parameters and values of sin2 (20) = 1.51 and Am 2 = 2.19xl0" 3 eV 2 ; ofsin 2 (20) = 1.00 and Am 2 = 2.79 x 1 0 - 3 eV2 if only results in the physical region are considered. The probability that the first, non-physical, result is obtained as a statistical

206

fluctuation if the true values are sin2 (29) = 1.00 and Am 2 = 2.79 x 10~ 3 eV2 is of 13%. This last set of parameters corresponds to a total number of events of 104.8 and an energy spectra shape, shown in Fig. 3, compatible in 36% with the one observed for 1^-ring events. Fits conducted using the two periods of K2K data separately or using the total number of events and the shape of energy spectra separately give results which are also compatible with the combined ones. At 90% confidence level these results are also compatible with the ones obtained for atmospheric neutrinos by the SuperKamiokande collaboration. The allowed regions for Am 2 and sin2 (29) are shown in Fig. 4 and more details on the analysis can be found in Ref. 12 . The K2K data favour an oscillation with maximal mixing and Am 2 G [1.9,3.6] x 10~ 3 eV2; based on likelihood ratio they exclude the null oscillation hypothesis at 4 ve oscillations. 4. Conclusions The K2K collaboration confirms, in a controlled way, the neutrino oscillations first seen in atmospheric events, excluding the null oscillation hypothesis at 4cr. K2K data favours an oscillation with maximal mixing and Am 2 G [1.9,3.6] x 1 0 - 3 eV 2 , at 90% confidence level.

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References 1. Super-Kamiokande Coll., Y. Ashie et al, Phys. Rev. Lett. 93, 101801 (2004). 2. K2K Coll., S. H. Ahn et al, Phys. Lett. B511, 178 (2001). 3. K2K MRD Group, T. Ishi et al, Nucl. Instrum. Meth. A482, 244 (2002) [Erratum-ibid. A488, 673 (2002)]. 4. K2K Coll., A. Suzuki et al, Nucl. Instrum. Meth. A453, 165 (2000). 5. K2K SciBar Group, K. Nitta et al, Nucl. Instrum. Meth. A535, 141 (2004). 6. K2K Coll., S. H. Ahn et al, Phys. Rev. Lett. 90, 041801 (2003). 7. Y. Hayato, Nucl. Phys. Proc. Suppl. 112, 171 (2002). 8. C. H. Lllewellyn Smith, Phys. Rept. 3, 261 (1972). 9. D. Rein and L. M. Sehgal, Annals Phys. 133, 79 (1981).

208 10. M. Gluck, E. Reya and A. Vogt, Z. Phys. C67, 433 (1995); A. Bodek and U. K. Yang, Nucl. Phys. Proc. Suppl. 112, 70 (2002). 11. D. Rein and L. M. Sehgal, Nucl. Phys.B223, 29 (1983); J. Marteau, J. Delorme and M. Ericson, Nucl. Instrum. Meth. A 4 5 1 , 76 (2000). 12. K2K Coll., E. Aliu et al, hep-ex/0411038, submitted to Phys. Rev. Lett.. 13. see http://harp.web.cern.ch/harp/, and CERN-SPSC/99-35 (1999). 14. see http://www-numi.fnal.gov/minwork/minoswk.html, and NuMI-L-337 (2000). 15. see http://proj-cngs.web.cern.ch/, and CERN AC/2000-03 (2000). 16. see http://neutrino.kek.jp/jhfnu/, and hep-ex/0106019 (2001).

SNO: SALT P H A S E RESULTS A N D N C D P H A S E STATUS

J. MANEIRA ON BEHALF OF THE SNO COLLABORATION. LIP-Lisboa Av. Elias Garcia, 14, 1- ; 1000-149 Lisboa, E-mail: [email protected]

Portugal

The Sudbury Neutrino Observatory (SNO) is an underground heavy-water Cherenkov detector designed to measure neutral (NC) and charged (CC) current interactions of 8 B solar neutrinos on deuterium. This talk focused on the characterization of the detector response and on the Physics results from the second phase of SNO, in which 2 tonnes of NaCl were added to the heavy water, in order to enhance the sensitivity to NC interactions. This allowed for precision, energyunconstrained measurements of the solar neutrino flux that confirmed solar model predictions and gave strong evidence for flavor change, excluding maximal flavor mixing at a level of 5
1. Introduction The discrepancy between measured 1 ' 2 ' 3 and predicted 4 solar neutrino fluxes, that lasted for over thirty years, was known as the solar neutrino problem. The solution was provided by the Sudbury Neutrino Observatory (SNO) results 7 , that showed that the i/e's produced in the Sun core undergo a flavor conversion on their way to the Earth. Neutrino oscillations 5 ' 6 are an elegant mechanism for this flavor conversion that is strongly supported by other experiments as well. SNO 9 is a heavy water Cherenkov detector located 2 km underground in a nickel mine near Sudbury, Canada. The active target of the detector consists of 1000 tonnes of heavy water(D20), contained in a 12 m diameter transparent acrylic vessel (AV) and observed by 9456 photomultiplier tubes (PMTs). The 8 B solar neutrinos are detected through the processes: ve+d->p + p + evx + d-+p-\-n-\-ux vx + e~ -> vx + e~

(CC), (NC), (ES).

While the charged current (CC) reaction is sensitive only to ve's, the

209

210

neutral current reaction (NC) is equally sensitive to all active neutrino flavors (a; = e,/x,-r), so its observation provides a model-independent measurement of the total flux of active 8 B solar neutrinos. The elastic scattering (ES) reaction is sensitive to all electron flavors as well, but with reduced sensitivity to v^ and vT. The NC neutron counting is carried out in different ways in each of the three phases of SNO: detection of a 6.25 MeV 7 ray following capture on a deuteron in the first phase; detection of a 8.6 MeV 7 ray cascade following capture on 35C1 in the second phase, where 2 tonnes of salt were added to the D2O. In the third phase, the neutrons will be detected by an array of 40 3 He counters (Neutral Current Detectors, or NCDs), independently from Cherenkov events. 2. The Salt Phase of SNO The salt data set presented here corresponds to 254.2 live days 8 . After removal of instrumental and non-Cherenkov like backgrounds, 3055 events were selected. The analysis used a kinetic energy threshold of T e / / > 5.5 MeV and a fiducial volume radius R/u < 550 cm. 2.1. Detector

Response

The times and positions of hit PMTs are used to calculate estimators for the vertex position, the event light isotropy and the particle direction and energy. The detector response was calibrated by using a source deployment system capable of reaching off-axis positions inside the acrylic vessel, as well as in vertical axes in the light water volume. The energy estimator uses an energy normalization obtained with the 16 N 7 source and an optical model calibrated with a laser diffusing source ("laserball") to estimate the event energy based on its number of hit PMTs (nhit), reconstructed position and direction. The laserball calibration measured an increase in optical attenuation in the D2O volume throughout the salt phase, as shown in Figure l(Left) and the 16 N calibration showed a steady drop in the detector response (about —2% yr~x). As shown in Figure 1 (Right), the drift was predicted by Monte Carlo simulations using time-varying D2O attenuation, so a correction was applied to the energy estimator. The uncertainty in the energy scale was measured with the 16 N calibration to be 1.1%. A 2 5 2 C / spontaneous fission source was used to calibrate the distribution of neutron detection efficiency in function of position in the detector.

211

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Figure 1. Optical and energy calibration. Left: Inverse attenuation length of D2O in function of wavelength, for different calibration periods in the salt phase, as measured with the laserball source. Right: Number of hit PMTs (corrected to the center of the detector) in function of date, for data (dark points) and Monte Carlo simulations(light points) as measured with the 1 6 N source.

The volume-weighted efficiency within the analysis window is 0.399±0.012, about a factor of three higher than in the pure D2O phase. The angular distributions of the PMT hit pattern of each event was expanded in terms of Legendre polynomials and the linear combination of coefficients ftu was chosen as the isotropy parameter. Single ring electron events are less isotropic than multiple ring 7 cascades, so /3i4 can be used to separate these classes of events. The /3i4 distributions were calibrated by comparing Monte Carlo to 2 5 2 C / and 16 N source data. The uncertainty in the mean value is 0.87%.

2.2.

Backgrounds

The total photo-disintegration backgrounds were estimated to be 113 ± 26 neutrons. The largest contribution is from uranium and thorium chain activity in the D2O and this was measured by ex-situ and in-situ methods throughout the data-taking period to be below the design goals. Background neutrons produced at the acrylic vessel or in the H2O can propagate into the fiducial volume, but the enhanced neutron capture efficiency in salt makes the external neutrons readily apparent, so an additional radial distribution function was included in the fits to extract the rate of this component. The contribution from low energy Cherenkov backgrounds was < 14.7 events (68% C.L.).

212

2.3.

Solar Neutrino

Results

The maximum likelihood method was applied to extract the fluxes of each neutrino interaction and the external neutron background rate from the salt data set. The likelihood function was constructed with probability density functions of volume-weighted vertex radius (p = (Rfit/RAv)3), event angle with respect to the Sun {cos6aun) and isotropy (Pu). As can be seen in Figure 2, the radial distribution allows the separation of backgrounds and (5u allows the separation of the CC and NC events, making possible an energy-unconstrained analysis. The fitted number of events yield the following fluxes, in units of 106CTO-2S-1:

$cc = i-59±8:8?(«tat.)±o:8i(sy»*0 $ B S = 2.2ltoil(stat.) $NC

±

O.W(syst)

= 5.21 ± 0.27(stat) ± 0.38(syst.)

The systematic uncertainties are listed in Table II of reference 8 .

Figure 2. Distribution of (a) /3i4, (b) volume-weighted radius for the selected events. Also shown are the Monte Carlo predictions for CC, ES, NC, internal and external neutrons scaled to the fit results. The dashed lines represent the summed components.

A combined x 2 analysis of the shape-unconstrained fluxes from the salt phase, the day and night energy spectra from the pure D2O phase and results from other solar neutrino experiments was carried in order to constrain the allowed regions of the neutrino mixing parameters Am 2 and tan20. When the results from the reactor neutrino experiment KamLAND 10 are included, the best fit point is Am 2 = 7.1+J^xlO - 5 , 9 = 32.5tlj, / B = 1-02 ( l a errors). At 99% C.L., only the lower band of the LMA region is allowed, and maximal mixing is disfavored at a level of 5.4o\

213 3. Status Report on the Third Phase of SNO The main goal of the third phase of SNO is the event-by-event measurement of the neutral current rate, independently from Cherenkov detection. The neutral current detector (NCD) array consists in 40 sets ("strings") of proportional counters filled with (4 of them with 4 He). The counters are built of low background nickel(<10 ppt U, Th). The strings have a diameter of 5 cm and a length of 9-11 m, depending on the attachment position in a l m x l m grid at the bottom of the AV. The capture of the neutron on 3 He yields a proton and a 3 H nucleus that can both be detected by ionization in the 3 He gas. The NCD electronics can acquire the full ionization waveform, that is used for the rejection of a//3 radioactivity background. As opposed to the first two phases of SNO, the NC and CC rate measurements in the NCD phase will be essentially independent. This will allow for a factor of 2 improvement in the precision of the CC/NC ratio and also improved 0\2 constraints, in a global oscillation analysis. After a one year period of NCD installation and commissioning, production data-taking for the third phase of SNO started in January 2005. Acknowledgments This work is supported by Canada: NSERC, Industry Canada, NRC, Northern Ontario Heritage Fund, Inco, AECL, Ontario Power Generation, HPCVL, CFI; USA: DOE; UK:PPARC; PortugakFCT. We thank the SNO technical staff for their strong contributions. References 1. B.T. Cleveland et al., Astrophys. J. 496, 505 (1998). 2. Y. Pukuda et al, Phys.Rev. Lett. 77, 1683 (1996); S. Fukuda et al, Phys. Lett. B539, 179 (2002) 3. V. Gavrin, Nucl.Phys.Proc.Suppl. 138, 87 (2005); W. Hampel et al., Phys. Lett. B447, 127 (1999); T. Kirsten, Nucl.Phys.Proc.Suppl. 118, 33 (2003) 4. J.N. Bahcall, A. Serenelli and S. Basu, Astrophys. J. 621, L85 (2005). 5. V. Gribov and B. Pontecorvo, Phys. Lett. B28, 493 (1969) 6. L. Wolfenstein, Phys. Rev. D17, 2369-2374 (1978); S.P. Mikheyev, A.Y. Smirnov, Sov. Jour. Nucl. Phys. 42, 913-917 (1985) 7. Q.R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett. 87, 071301 (2001); Phys. Rev. Lett. 89, 011301 (2002); Phys. Rev. Lett. 89, 011302 (2002). 8. S.N. Ahmed et al. (SNO Collaboration), Phys. Rev. Lett. 92, 181301 (2004); New paper on the full dataset submitted after the Workshop; nucl-ex/0502021 9. SNO Collaboration, Nucl. Instr. and Meth. A449, 172 (2000). 10. K. Eguchi et al, Phys. Rev. Lett. 90, 021902(2003)

T H E ICARUS E X P E R I M E N T

S. NAVAS-CONCHA* Dpto.

de Fisica

Tedrica y del Cosmos & C.A.F.P.E., University Av. Severo Ochoa s/n, 18071 Granada, Spain E-mail: [email protected]

of

Granada,

The ICARUS detector is a liquid argon time projection chamber. It provides three dimensional imaging and calorimetry of ionizing particles over a large volume, with high granularity. This multipurpose detector opens up unique opportunities to look for phenomena beyond the Standard Model through the study of atmospheric, solar and supernova neutrinos, nucleoli decay searches and neutrinos from the CERN to Gran Sasso beam. The ICARUS technology has reached maturity with the construction and test (during summer 2001) of a 600 ton detector, demonstrating the feasibility of building large mass devices relevant for non-accelerator physics. The 600 ton detector was transported to the Gran Sasso laboratory (Italy) in December 2004. A summary of the most relevant results obtained during the test period as well as an overview of the general physics program of the ICARUS experiment are reported.

1. The ICARUS T600 detector The ICARUS T600 liquid Argon (LAr) detector 1 consists of a large cryostat split in two identical, adjacent half-modules, each of 3.6 x 3.9 x 19.9 m 3 internal dimensions. Each half-module is an independent unit housing an internal detector composed by two Time Projection Chambers (TPC), a field shaping system, monitors and probes, and by two arrays of photomultipliers. Externally the cryostat is surrounded by a set of thermal insulation layers. The TPC wire read-out electronics is located on the top side of the cryostat. The detector layout is completed by a cryogenic plant made of a liquid Nitrogen cooling circuit to maintain uniform the LAr temperature, and of a system of LAr purifiers. A liquid Argon TPC allows to detect the ionization charge released at the passage of charged particles in the volume of LAr, for three dimensional image reconstruction and calorimetric measurement of ionizing events. The •On behalf of the ICAURS Collaboration.

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detector, equipped with an electronic read-out system, works as an "electronic bubble chamber" employing LAr as ionization medium. Unlike traditional bubble chambers, limited by a short window of sensitivity after expansion, the LAr TPC detector remains fully and continuously sensitive, self-triggerable and without read-out dead time. A uniform electric field applied to the medium makes the ionization electrons drift onto the anode, following the electric field lines; thanks to the low transverse diffusion of the ionization charge, the electron images of ionizing tracks are preserved. Successive anode wire planes, biased at a different potential and oriented at different angles, make possible the three dimensional reconstruction of the track image. The wire pitch is 3 mm. The maximum drift path (distance between the cathode and the wire planes) is 1.5 m and the nominal drift field 500 V/cm (a 3 meter drift is foreseen for future ICARUS modules). Each wire of the chamber is independently digitized every 400 ns. The electronics was designed to allow continuous read-out, digitization and independent waveform recording of signals from each wire of the TPC. Measurement of the time when the ionizing event occurred (so called "To time" of the event), together with the electron drift velocity information, provides the absolute position of the tracks along the drift coordinate. The TQ can be determined by detection of the prompt scintillation light produced by ionizing particles in LAr 2 .

2. The T600 t e s t run A full above-ground test of the T600 experimental set-up was carried out in Pavia (Italy) during summer 2001. One T600 half-module was fully instrumented to allow a complete test in real experimental conditions. All technical aspects of the system, cryogenics, mechanics, LAr purification, read-out chambers, scintillation light detection, electronics, and DAQ were tested, and found to be satisfactorily in agreement with expectations. During the test run, a very large amount of cosmic ray events was recorded 3 with different configurations of a dedicated trigger system: long, penetrating muons and spectacular, high multiplicity muon bundles, electromagnetic and hadronic showers and low energy events among other categories of events. Like a bubble chamber, the ICARUS detector provides a measurement of the total ionization loss of a track with very high sampling. By extracting the physical information contained in the wires output signal, i.e. the energy

216 deposited by the different particles and the point where such a deposition has occurred, it is possible to build a complete three dimensional spatial and calorimetric picture of the event. The measurement of the dE/dx and positions for a large number of points along a given track provides a way of estimating the particle momentum -from range (for stopping particles) or multiple scattering, providing, in addition, a method for particle identification. As an example, Figure 1 shows a kaon decay candidate event acquired during the T600 test.

Figure 1. Charged Kaon decay candidate from the T600 test run. The Kaon, muon and electron are visible from right to left.

A fundamental requirement for the ICARUS technology is that electrons produced by ionizing particles might travel unperturbed in LAr from the point of production to the collecting wire planes. To this extend, the concentration of any kind of electro-negative impurity diluted in the liquid must be reduced at extraordinarily low levels. At the end of the run, the measurements provided by the Purity Monitors (dedicated devices immersed in the LAr) and from crossing muon tracks shown a maximum value of the drift electron lifetime of about 1.8 ms, equivalent to an electron mean free path greater than 280 cm, and was still increasing 4 (see Figure 2). Another example that proves the ICARUS detector quality is given through the measurement of the muon decay energy spectrum from a sample of stopping muon events. The Michel p parameter was measured 5 by comparison of the experimental and Monte Carlo simulated /x spectra. The obtained value p = 0.72 ± 0.06 ± 0.08 is in agreement with the SM value p = 0.75. The data coming from the test run provided also the possibility to per-

217 ,—> 2500 g

2250

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10

20

30

ICARUS T600 ST

§

40

50

60

70

80

Elapsed Time (Days) Figure 2.

Evolution of the free electron lifetime during the T600 run.

form a precise measurement of the electron recombinacion in liquid Argon 6 . This parameter, crucial for the proper determination of the energy released by the particles in the medium, was measured by means of charged particle tracks. The values found for the normalization factor and the Birks law parameters are compatible with other liquid Argon TPC prototypes and other data published in the literature. 3. The ICARUS physics program A liquid Argon time projection chamber (TPC), working as a electronic bubble chamber, continuously sensitive, self-triggering, with the ability to provide 3D imaging of any ionizing event, together with excellent calorimetric response, offers the possibility to perform complementary and simultaneous measurements of neutrinos, as those of the CNGS beam, those

218 from cosmic ray events, and even those from the sun and from supernovae. The same class of detector can also be envisaged for high precision measurements at a neutrino factory 7 ' 8 and can be used to perform background-free searches for nucleon decays. Hence an extremely rich and broad physics program, encompassing both accelerator and non-accelerator physics, will be addressed. These will answer fundamental questions about neutrino properties and about the possible physics of the nucleon decay. The T600 detector was transported underground inside the Gran Sasso laboratory in December 2004. The first data is expected to be taken by spring 2005. Though it has a physics program on its own 9 , the T600 construction was mainly motivated by technical issues. Since the concept of modularity was inserted starting from the very initial phases of the detector design, growing the detector mass to the several kton scale can be accomplished by "cloning" the actual T600 to the required number of units. The construction of the first 1200 ton module will start this year. Acknowledgments The author wishes to thank the conference organizers for preparing such an excellent workshop. The credit of the work presented on this report is due to the ICARUS Collaboration members that contributed to the detector construction, operation and physics analysis. The author gratefully acknowledges support from the Spanish Ministry of Education and Science and the Ramon y Cajal Programme. References 1. S. Amerio et al. [ICARUS Collab.], Nucl. Instrum. Meth. A527, 329 (2004). 2. M. Antonello et al. [ICARUS Collab.], Nucl. Instrum. Meth. A516, 348 (2004). 3. F. Arneodo et al. [ICARUS Collab.], Nucl. Instrum. Meth. A508, 287 (2003) [Erratum-ibid. A516, 610 (2004)]. 4. S. Amoruso et al. [ICARUS Collab.], Nucl. Instrum. Meth. A516, 68 (2004). 5. S. Amoruso et al. [ICARUS Collab.], Eur. Phys. J. C33, 233 (2004). 6. S. Amoruso et al. [ICARUS Collab.], Nucl. Instrum. Meth. A523, 275 (2004). 7. A. Bueno, M. Campanelli and A. Rubbia, Nucl. Phys. B589, 577 (2000). 8. A. Bueno, M. Campanelli, S. Navas-Concha and A. Rubbia, Nucl. Phys. B631, 239 (2002). 9. F. Arneodo et al. [ICARUS Collab.], LNGS-P28/2001 (2001).

Cosmological Parameters Measurements

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HIGH R E D S H I F T S U P E R N O V A SURVEYS

SEBASTIEN FABBRO* CENTRA - 1ST, Avenida Rovisco Pais 1049-001 Lisbon, Portugal E-mail: [email protected]

Few years ago, supernova observations have demonstrated their essential role in cosmology, by providing direct evidence for a dark energy component, responsible for the acceleration of the universe. We review how type la supernovae are used as distance indicators for cosmological parameter studies, and present the attempts of current surveys to turn pioneering distant supernova discoveries into daily routine, in order to provide a direct measurement of the dark energy equation of state.

1. Introduction The quest for cosmological parameters is tightly linked to measuring distances. The greater the distances are, the more they give access to the geometry and the evolution of the universe. Since the first Hubble diagram in 1929, there has been several attempts to get good cosmological distance indicators. One type of astronomical source has been recently successful in converting redshifts to Megaparsecs with enough accuracy: Type la Supernovae (SNIa). Two observations are needed to be able to use SNIa as cosmological distance indicators. From spectra we measure the redshift z and identify the supernova, and from multi-band photometry we reconstruct the light curve F(t) of the SNIa. Relating the two observations is done through the luminosity distance d^ derived directly from the inverse-squared relation F(t) = L(t)/4nd^(z;p), where L(t) is the intrinsic luminosity and F(t) the observed flux. The cosmological parameters p are only included in dj_,, supernovae observations are therefore "cosmological-model-independent". In a standard concordance *Work supported by project POCTI/CTE-AST/57664/2004 of the Fundacao Cienca e Tecnologia.

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cosmological model, the luminosity distance relates with the reduced matter energy density : fim, the reduced dark energy density fix, and the ratio of pressure and density of the dark energy component w(z) = p/c2p(z). For a classic cosmological constant, we have simply w(z) = —1, but many other possibility1 are speculated given the little observational constraints we have on w(z). In practice, we need also to constrain hL(t). This is done with nearby SNIa, and the same objects also allow corrections for real differences found in their light curves. Fortunately, these corrections are tractable: at a first order, the brighter objects are slower and bluer, and a semi-parameterized template can be constructed 2 to match pretty much all SNIa observations, and reduce scatter in the distance estimators down to 7%.

2. Requirements for a Supernova Survey How well are we able to determine energy densities with the SNIa Hubble diagram? Apart from observational systematics and a SNIa intrinsic dispersion of 0.15mag, the precision on the energy densities and equation of state are limited. Energy densities and equation of state evolution parameters are

2.0 1.5 1.0 0.5 4

0.0

-0.5 -1.0 -1.5 "2'8.
0.5

1.0 z

1.5

2.0

Figure 1. Distance modulus partial derivatives with respect to some cosmological parameters, in function of redshift.

223

strongly degenerated through any cosmological distance relations. Given a cosmological parameter set p = {h,flm,Qx,wo,wa}, where we chose a simple equation of state redshift evolution w(z) = w0 + waz/(l + z), we can compute a the Fisher matrix, which under reasonable assumptions, depends only on luminosity distance derivatives. In Figure 1, we show the derivatives of the distance modulus fi = 51og(c^(2:;p)/10pc) with respect to different cosmological parameters, with a fiducial set of cosmological parameters p = {0.7,0.3,0.7, —0.8, —0.4}. The more similar the curve shapes are, the harder it is to break the degeneracy between the corresponding cosmological parameters. Supernovae observations (as well as any other observations so far) are therefore not self-sufficient to determine with decent accuracy the full set of cosmological parameters. The situation is different for other parameterization of the equation of state or another set of fiducial values for the model. If the dark energy sector is only populated by a pure positive cosmological constant, the supernovae observations alone remain very strong to constrain cosmological parameters. As noted previously by several authors 3 , the sensitive redshift region for a joint estimation of Clm and f2x is about z ~ 0.7. Even though distance measurements have their limits to trace a full history of cosmological parameters, supernova surveys are very effective to determine dark energy density and its evolution up to redshifts of 2. To allow constraints on the density parameters, we will then have to either combine other experiments, or assume priors on the parameters. Typical assumptions are a flat universe, justified e.g. by cosmic microwave background results 4 , and a value of Qm which can be measured better than 10% with galaxy surveys 5 . To also achieve 10% on fix and wo, we need the good statistics of ongoing and future supernova surveys to beat the systematics resulting mostly from the current supernovae intrinsic dispersion. Obtaining a set of supernovae at various redshift is a dedicated task, on which two teams have concentrated their efforts for the last decade. SNIa only appear few times per millennium per galaxy and can be photo-metered for about 200 days in rest frame at low-z. Supernova surveys require both a wide field and a large mirror to cover a large volume of space, getting millions of galaxies. A discovery strategy has been set up through the nineties by the two collaborations, leading to the observational evidence of an accelerating universe driven by a repulsive component 6 ' 7 . The use of 4-m class telescopes with the appearance of high quantum efficiency CCDs allowed such discoveries. Since a SNIa rises in about ~ 20 days from undetectable light

224

to maximum light, the strategy elaborated was to use a 3 weeks baseline between moonless reference images and discovery images of the same fields, and look for SNIa candidates in the subtracted images, using fine-tuned image processing software. Candidates were whereupon confirmed spectroscopically in 8m-10m class telescopes and followed-up photometrically in as many telescopes as allocated to build the light curves. Today, this strategy does not hold up with the needs for greater precision and the pressure on telescope allocations. We will see in the next sections, current and future supernova projects elaborated to get much higher statistics. 3. Ground Surveys The SuperNova Legacy Survey (SNLS, http://www.cfht.hawaii.edu/SNLS) is one of the Canada France Hawaii Telescope (CFHT) Legacy Survey that started in 2003. It uses the Megaprime imager, a wide-field camera (Megacam) mounted on the prime focus of the CFHT-3.6m telescope in Hawaii. The camera is a mosaic of 36 thinned 2Kx4.5K CCDs covering one square degree. Four fields are observed in four bands (gM,rM, iht, ZM), in average 18 nights centered on each new moon, until 2008. The observation strategy allows early discovery and well-sampled homogeneous follow-up photometry of candidates, since images are used by both online detection and follow-up photometry. To reach redshifts of z ~ 1, where the supernova reaches faint

Figure 2. Left: SNIa redshifts acquired up to April 2004, with the SNLS. Right: a sample multi-color griz SNIa light curve at z = 0.91 obtained with SNLS offline photometry.

magnitudes such as i = 24.5, 8-10 meter class telescopes are needed for spectroscopy. Spectroscopic follow-up has been acquired so far at the Eu-

225

ropean Southern Observatory Very Large Telescope 8m, Gemini North and South 8m telescopes, as well as the Keck 10m telescopes. Accumulation of supernovae is shown in Figure 2, with a sample light-curve. If the SNLS continues on current tracks it already started, it could very well satisfy its objectives of cosmological constraints as shown in Figure 3.

with a(£2M)=0.03

with a(Q M H>.015

0.2

n„

a„

0.4

nu

Figure 3. Expected fim - wo 39%, 86%, and 99% CL contours after 5 years of SNLS survey and 200 nearby SNIa. The fiducial cosmological model was for a cosmological constant and Q m = 0.3, and the contours account for photometric errors and a 0.15 mag intrinsic dispersion.

The Equation of State: SupErNovae trace Cosmic Expansion (ESSENCE) project is a similar project in the Southern hemisphere, at the Cerro-Tololo 4-m telescope, It started in 2002 and aims to observe well 200 SNIa between in the redshift range 0.15 - 0.75, in VRI. Supernovae at higher redshifts are expected to be followed with the Hubble Space Telescope (HST), and redder pass-bands photometric data acquired with other ground-based telescopes. More information on the project can be found on http://www.ctio.noao.edu/~wsne. 4. Space Surveys For supernovae at redshift z > 1, needed to increase the leverage and beat the 5% measurements of cosmological parameters, space observations are required. Only space observations can give us good photometric precision and spectroscopy at higher redshift, where most of the SNIa flux is in infrared. There are two dependent program using currently the HST with the AFS camera: the Supernova Cosmology Project (SCP) has just completed a first search using HST-ACS (see http://supernova.lbl.gov) and the

226

PANS-GOODS survey, which has already published an analysis with 16 new supernovae followed with HST-ACS 8 , and continue similar efforts. With the help of a supernova compilation of other surveys, the PANS-GOODS survey were able to get a first measurement of the epoch of deceleration. The SuperNova Acceleration Probe (SNAP) is a project of a 2-m telescope in space, mounted with a wide field imager, an infrared imager and an integral field spectrometer. The observation strategy is similar to the SNLS, but covering 15 square degrees and reaching 2000 well followed SNIa/year, with an expected photometric precision of less than 2%. SNAP data quality should be able to discriminate among dark energy models and alternative explanations to the acceleration of universe expansion. In principle, it would be able to detect time variation of the equation of state, together with prior information on flm and low-redshift SNIa from the Nearby Supernova Factory. The SNAP design is still evolving. Depend-

0.5

i

•

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'

'

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SNAP I SNAPrfNF

0.3 0.2

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-1

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Figure 4. Expected confidence contours (68%) in the (wo,u;i) plane for the SNAP experiment when the Nearby Supernova Factory SNIa are added (in red-dashed) and when they are not. A flat universe has been assumed with a Gaussian prior on f2 m of
ing on the availability of the funds, SNAP is expected to be launched in 2010 and its mission should last at least for three years. It is also designed to do together with supernovae, weak lensing observations, therefore nicely complementing the dark energy measurement.

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5. Conclusion The golden age of supernova searches has started. We should be seeing wagons of high redshift supernovae data publications in the coming years, thanks to ground redshift large surveys as SNLS or ESSENCE. Space surveys can not yet be as productive with only HST, but each higher redshift SNIa brings more cosmological information. Future space surveys could bring even more. But to guaranty success in all these projects, we need a better supernova understanding. The Nearby Supernova Factory Project, described in this same volume 9 , has been designed to satisfy this need. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Padmanabhan, T., Physics Reports, 380, 235 (2003) Guy, J. et al., astro-ph/0506583 (2005) Astier, P., Physics Letters B 500, 8 (2001) Spergel, D. et al., Astrophysical Journal Supplement 148, 17 (2003) Eisenstein, D. et al., astro-ph/0501171 (2005) Perlmutter, S. et al, Astrophysical Journal 517, 565 (1999) Riess, A. et al., Astronomical Journalll6, 1009 (1998) Riess, A. et al., Astrophysical Journal 607, 665 (2004) Antilogus, P. this volume.

SNFACTORY : N E A R B Y S U P E R N O V A FACTORY

P. ANTILOGUS The Nearby Supernova Factory: P. Antilogus, G. Garavini, S. Gilles, L-A. Guevara, D. Imbault, R. Pain, D. Vincent (LPNHE, Paris), G. Adam, R. Bacon, C. Bonnaud, L. Capoani, D. Dubet, F. Henault, B. Lantz, J-P. Lemonnier, A. Pecontal, E. Pecontal (CRAL, Lyon), N. Blanc, G. Boudoul, S. Bongard, A. Castera, Y. Copin, E. Gangler, G. Smadja (IPNL, Lyon), R. Kessler (KICP, Chicago), G. Aldering, B. C. Lee, S. Loken, P. Nugent, S. Perlmutter, J. Siegrist, R. Scalzo, R. C. Thomas, L. Wang (LBNL, Berkeley), C. Baltay, D. Rabinowitz, A. Bauer (Yale) E-mail: [email protected] The Nearby Supernova Factory (SNfactory) is an experiment designed to collect a large sample (300 or more) of low redshift ( 0.03 < z < 0.08) Type la supernovae, each followed up by 10-15 spectro-photometry (320-1000 nm) observations spaced at roughly 3 day intervals, and started ~ 10 days before the maximum light. The goal of this experiment is to anchor the supernova Hubble diagram at low redshift and to provide a large sample of well studied SN la to determine those properties of SNe la affecting their use for cosmology. The status of the SNfactory experiment will be presented, including informations on the first spectro-photometric data collected by SNIFS, the dedicated Integral Field Unit (IFU) for SNe observation.

1. Introduction The SNfactory project has been initiated by scientist at LBNL (Berkeley), KICP (Chicago), Yale in the United States; CRAL (Lyon), IPNL (Lyon) and LPNHE (Paris) in France. The SNfactory will concentrate on Type la supernovae (SNe la), the type which has been used to determine that the expansion of the universe is accelerating x. The SNfactory will lay the foundation for the new generation of experiments (SNLS ,ESSENCE) to measure the expansion history of the Universe. It will discover and obtain light curve of spectrophotometry quality for ~ 300 Type la supernovae in the low-redshift end of the smooth Hubble flow.

228

229

-14 days''Z

>,

> +10 day
!

r?r

>

+20 days--

/ /,

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Figure 1. As the ashes of the SN -5=4. la explosion expand with time, the light collected is sensitive to the composition (spectral features) and the speed (width of the spectral features) of deeper and deeper layers of the SN la. By observing such time series the 'J 1, SNfactory project will improve our understanding of the SN la explosion.

2. Science goals The SNfactory science goals are (see Fig. 1 and Ref. 2 for details ) : • to anchor the low red-shift SN la hubble diagram : the minimization of the errors on the cosmological parameters by the highredshift SNe experiments, such as SNLS or ESSENCE, can be improved by a factor ~ 2 using the unique large sample of SN la expected by SNfactory, • to calibrate/measure technical quantities needed for the analysis of high red-shift SNe la (stretch-luminosity relation, K-corrections, intrinsic colors, etc.) • to improve the quality and the understanding of the SNe la as distance indicator.

3. The SNfactory

project

The supernova discovery for SNfactory is being carried out by the PalomarQUEST project using the 1.2m aperture Samuel Oschin Schmidt Telescope

230 Red channel irucro-iens array

x cnanne s (blue + red channels)

1 spectra per lens =225 spectra per channel

Sum the lenses/spectra with Ugh: from the SN

Figure 2. SNIFS principle: the SN and surrounding galaxy is sampled by two 15 x 15 micro-lens arrays (one in the blue , one in the red) covering a field of 6" x 6". A spectra is produced with the light collected by each lenses. From the data cube produced ( = measured intensity for a given position in the field of view and a given wave length ), a photometric spectra of the SN la is extracted. It should be noticed than an IFU allows spectro-photometric measurements. This is not the case of spectra obtained with a simple slit as the fraction of the spectered object masked by the slit is unknown. The SN la spectra obtained at different epochs, integrated in a given wave length domain, allows to produce the SN la light curves in any filter.

at the Palomar Observatory with the large area QUEST camera a built at Yale and Indiana Universities. Traditionally supernovae have been followed with BVRI photometry, and spectra beyond the initial confirmation spectrum are rare. The SNfactory changes all that. Using a integral field unit 3 on a two-channel (blue 3200-5400A& red 5000-10000A) optical spectrograph with a resolution of respectively 4.6 A and 6 A, the SNfactory's SuperNova Integral Field Sspectrograph (SNIFS, build in Prance) allows spectroscopy of supernovae at all epochs. Because these spectra are spectrophotometric, UBVRIZ photometry can be synthesized from these spectra, without the uncertainties due to photometric color terms and K-corrections (see Fig. 2). SNIFS rea

The QUEST camera covers 2.3° x 4.0° with 112 CCD of 2.4k x 0.6k.

231 tains one advantage of the traditional approach, which allows surrounding field stars to be used for flux scaling when conditions are non-photometric, by also having an imager which integrates on the field immediately surrounding each supernova and having the exact same exposure as the integral field unit. The follow-up for the SNfactory is performed at the University of Hawaii's 2.2-m telescope, located on the summit of Mauna Kea. Mauna Kea has excellent atmospheric stability producing some of the best image quality attainable from the ground. The SNfactory has arranged for observation with SNIFS 20% of the time on the 2.2-m, allocated as the 2nd half of the night on every first and third night of a five-night cycle. The SNfactory program plan to collect 300 nearby SN la ( 0.03 < z < 0.08 ) and to do a detailed study of the spectral evolution of each SN la: 10-15 spectra of spectro-photometric quality between 15 days before and 45 days after maximum. 4. First observations with

SNIFS

The dedicated SNIFS IFU has been build in 2002-2003 and mounted on the UH 2.2m telecope in Hawaii in March 2004. SNIFS got its first light in April 2004, and was fully operational in July 2004. Since then, it is controlled remotely and has an automated target acquisition using directly the finding charts made with the discovery image. SNfactory science program started in August 2004 with a test run using supernova discovered by other observers. End of June 2005, the search at Palomar, which had been successfully tested in 2002 with a smaller camera, will provide a regular sample of la to follow with SNIFS. Since August 2004, SNIFS has been collecting data 1/2 night 3 times a weak, each week. During this first test run (mid-August 2004 - endDecember 2005) SNIFS has followed 5 SN : • • • • •

SN2004dt , SN la : 29 observations (4 Months, z=0.02 ) SN2004ef , SN la : 13 observations (2 Months, z=0.03 ) SDSS-SN191 , SN la : 3 observations (1 week, z=0.1 ) SN2004gk , SN Ic : 7 observations (1 Month, z=-0.0004 ) SN2004gs , SN la : 8 observations (4 weeks, z=0.027)

It has also screened 20 other SN, getting 1 spectra for each of them for identification (II,Ic/b,Ia). The current SNIFS efficiency allows to follow 4-5 SN la each 1/2 night. An improvement up to factor 2 is foreseen, after implementing the hardware

232

fixes and software changes than the test run has underlined. Calibrating a given standard star by an other one collected the same photometric night, has shown photometric measurement at the 1% level between 320-850 nm. Work is underway to perform photometric calibration in non-photometric night using the photometric channel, and to improve the photometry above 850 nm, limited at the moment by fringing on the CCD. 5. Conclusion The SNIFS IFU is fully operational and has started to take data. First spectro-photometry time series of SN la have been obtained. During the summer 2005, the SNIFS instrument will start to follow SN la provided by the search pipeline based on the QUEST camera data at Palomar. At the fall 2005 SNIFS should follow 5 SN la in parallel, corresponding to one new SN la per week and a rate of 50 SN la time series per year. References 1. S. Perlmutter et al., Ap. J. 517, 565 (1999). 2. G. Aldering et al., Proceeding.SPIE 4836, 61 (2002). 3. R. Bacon et al, A&AS 113, 347 (1995).

A POLARIZED GALACTIC EMISSION MAPPING EXPERIMENT AT 5-10 GHZ DOMINGOS BARBOSAf Centro de Fisica, Universidade do Porto, Rua do Campo Alegre 687 Porto, 4169-007, Portugal & CENTRA, Instituto Superior Tecnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal RUIFONSECA Instituto de Telecomunicacoes, Campus Universitario de Santiago, 3810-193 Aveiro, Portugal & CENTRA, Instituto Superior Tecnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal DINIS M. DOS SANTOS Dep. Electronica e Telecomunicacoes, Universidade de Aveiro 3810-193, Aveiro, Portugal LUIS CUPIDO Centro de Fusao Nuclear, Instituto Superior Tecnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal ANA MOURAO CENTRA, Instituto Superior Tecnico, Av. Rovisco Pais 1049-001, Lisboa, Portugal GEORGE F. SMOOT Astrophysics Group, Lawrence Berkeley National Laboratory, 1 Cyclotron Road 94720 Berkeley, USA & Physics Dept. , University of California, 366 Le Conte Hall, 94720 Berkeley, USA CAMILO TELLO Instituto Nacional de Astrofisica Espacial, Div. Astrofisica, Caixa Postal 515 12210-070 Sao Jose dos Campos SP, Brazil

Email: [email protected]

233

234 We show in this paper the goals of the Galactic Emission Mapping (GEM) project at Portugal. This project aims to map the North Hemisphere low frequency polarized foreground contaminants of the Cosmic Microwave Background in the frequency range 5-10GHz. The resulting maps will be merged with a correspondent map of the southern Hemisphere obtained by the GEM-Brazil team to produce an almost all-sky template of the polarized synchrotron. This data will be important for proper polarized foreground extraction in the data processing of Planck Surveyor and other near-future Cosmic Microwave Background polarization B-mode probing experiments.

1. Towards CMB polarization measurements Since 1992 [1], Cosmic Microwave Background (CMB) cosmology made a huge leap forward towards the era of high-precision measurements. The highresolution map data returned from experiments like MAXIMA, Boomerang, DASI, NASA's WMAP [3-6] offer a direct glimpse into the physics at the surface of last scattering. These data provide strong constraints on cosmological parameters and theories of large scale structure formation and favors the inflationary paradigm. Recently, the predicted low level of CMB polarization (E-modes) was detected (DASI - 2002, WMAP - 2003) [7-9] and will constitute for the next decade the best probe of early Universe's physics. It will also offer insights on Large-Scale structure formation by measuring the effects of reionization produced by the first galaxies. Besides providing a consistency check on the inflationary scenario favored by observations, the great interest in polarization measurements comes with the recognition that CMB polarization carries the imprint of the primordial gravitational background (large scale Bmode). The same inflationary mechanism responsible for the production of the small matter density fluctuations leading much latter to the Large Scale Structure, also produce a stochastic background of gravity waves (see Zaldarriaga [2] and references therein) . Yet, our view of the CMB is obscured by extragalactic and galactic foregrounds. Typically diffuse galactic emission is dominated by synchrotron radiation below 60GHz and by thermal dust emission above 60GHz [15]. Although good foreground templates exist for the main temperature anisotropies contaminants, namely galactic synchrotron emission and dust emission from galactic dust cirrus, little is known from templates about the polarized foregrounds. 2. The GEM project @ Portugal Experiments like Planck Surveyor will require well estimated templates of polarized synchrotron (affecting the Planck Low Frequency Instrument 30-90 GHz) and polarized dust emission (affecting the Planck High Frequency Instrument 100-800 GHz). Existing polarization templates (for low frequencies synchrotron) rely on a semi-empirical approach of extrapolating polarization substructure information from small sky area surveys with theoretical assumptions of depolarization of the synchrotron observed templates [10,13,14].

235 Furthermore, correlations of the WMAP first-year data with available foreground templates show another physical component appearing to be important - spinning dust emission which was not accounted for in the data processing analysis [17]. Thus, with foreground estimation as one of the main tasks currently being pursued by CMB teams, the Galactic Emission Mapper1 GEM synchrotron coverage simulation Stokes I: 5,0GHz Pormg&l Site (kmg- fl.I? ]at- r">.1.Vj

Q.QQ22 m^*®mmm^^mmmmmmmmm«««

Jwhui- 30 arc>mr.

0.1? K

(GEM) - a portable 5.5-m dish originally aiming to determine the spatial distribution and absolute intensity in the radio and microwave spectrum of the radiation emitted by the Milky Way galaxy [11,12] - was upgraded towards polarized foreground cartography at 5-10GHz (C. Tello, in preparation) with the development of a new high-sensitivity pseudo-correlator receiver. It will map the polarized synchrotron radiation (polarized at a maximum of 60% level at 10 GHz) where this emission is dominant. This dish is currently operated in Cachoeira Paulista, Brazil and will map I, Q, U Stokes parameters in. the southern hemisphere sky down to a lmK sensitivity at '5GHz, Recently, a new member was added to the GEM project (Barbosa et al.9 in preparation): seeking complementarity to the very near-future GEM-Brazil maps, a new dish (9-m) equipped with a high-sensitivity polarizer will operate in central Portugal with similar scanning strategy (steady, slow azimuthal rotation at fixed elevation at Irpm until the required sensitivity is achieved) aiming the coverage of the northern .hemisphere at the same frequencies at a latitude of -40° allowing a good coverage of high-galactic latitudes [17] (see Fig.l), target of future polarization .B-raode .probing experiments.

USA site: http://aether.Ibl.gov/www/proiects/gem Brazil site: http://www.cea.inpe.br/~cosmo/GEM/inciex_gem.htm

236 GEM synchrotron coverage simulation Stokes U: 5.0GHz Portugal Slt-s flong- fl.17 tat- 38 13)

fwhm« 30 arena.™

2„5«—0a m - m a s s a ^ m s f c « . * ^ s « » * ! « ^ -

v « O.fHXi K

Figure 1. Maps of the I, Q, U Stokes parameters of synchrotron galactic radiation to be obtained at 5GHz in central Portugal with a resolution of 30 arcmin. The simulations used 30GHz templates [13] scaled down to 5GHz. The simulated scanning strategy was azimuthal rotation at Irpm at a fixed elevation of 50°. It would gather information on the ISQ,U Stokes parameters in the northern sky after 6 months of integration time over 40% of the sky with a- resolution of 30 arcmin (the. signal to detect is -1 mK). At the same time, continuous work on the receiver will prepare the upgrades towards 10GHz observations, where some signal from spinning dust is expected to appear mixed to synchrotron emission. At a later stage, by 2007, GEM-Brazil and GBM-Portugal maps will be merged to produce the biggest sky templates of the polarized synchrotron components (-80% sky). These templates will shed light on the synchrotron polarizationfractionand substructure at scales down to 0.5°, covering a range of scales important for B-mode searching and allowing a better polarized foreground subtraction from large sky data sets such as the Planck Surveyor maps. Acknowledgments This work was supported by Funda?Io para a Ciincia e Tecnologia - Portugal through projects POCTI/FNU/42263/2001 and POCI/CTE-AST/57209/2004. DB is supported by a rolling CFP grant. RF is supported by a BIC grantfromIT - pdlo Aveiro.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

G. F. Smoot et al, Astrophy. J. 369, LI (1992). M. Zaldarriaga, Carnegie Obs. S., vol.2, (2003), astro-ph/0305272. P. de Bernardis, Nature. 404, 955 (2001). A.T. Lee et al. Astroph. Jour., 561, L1-L5 (2001). N.H. Halverson et al, Astrophys.J.. 568, 38-45 (2002). C.L. Bennet et al, Astroph. Journ. Sup., 148, 1 (2003). E.M. Leitch et al, Nature. 420, 763 (2002). J. Kovac et al., Nature. 420, 772 (2002). A. Kogut et al., Astroph. Journ. Sup, 148, 161 (2003). A. de Oliveira-Costa etal., New Astron. Rev., 47, 1117 (2003). C. Tello et al., Astronomy & Astrophys., 145, 495 (2000). S. Torres et al., Astroph. Space Sci., 240, 225 (1996). G. Girardino et al., Astronomy & Astroph., 387, 82, (2002). C. Baccigalupi et al., Astronomy & Astrophys. 372, 8 (2001). M. Tegmark et al., Astroph. J, 530, 133-165 (1999). D. Finkbeiner, Astrophys.J, 614, 186-193 (2004). R. Fonseca et al.,New Astron. (submited), astro-ph/0411477 (2004).

GALAXY CLUSTERS AS PROBES OF DARK ENERGY* P. T . P. V I A N A Departamento de Matemdtica Aplicada, Faculdade de Ciencias da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal Centra

de Astrofisica da Universidade Rua das Estrelas, 4150-762 Porto, Portugal E-mail: [email protected]

do

Porto

Clusters of galaxies are the most massive virialized structures in the Universe. Given that the mass function of large-scale structures decreases exponentially at the high-mass end, galaxy clusters are a sensitive probe of its normalization and redshift evolution, and hence of the cosmological parameters that most influence it. It will be discussed to what extent the present amplitude of density perturbations, the matter density, and the density and equation of state of the dark energy, can be constrained using observational data on the present and past abundance of galaxy clusters. Current results will be reviewed, and the expected constraints from the XMM-Newton Cluster Survey (XCS) will be presented.

1. Introduction Clusters of galaxies are one of the most important probes of the large scale structure and overall dynamical state of the Universe. Their present-day average statistical properties, and their evolution as a function of redshift, can be used to constrain the cosmological parameters that most influence the formation and evolution of large scale structures: the normalization of the power spectrum of density fluctuations, usually given as as - the dispersion of the density field on scales of 8 h"1 Mpc (h is the present value of the Hubble parameter, Ho, in units of 100 k m s - 1 M p c - 1 ) ; the total matter density in units of the critical density, fim; the energy density associated *Work partially supported by grant POCTI/FNU/43753/2001 of Fundagao para a Ciencia e a Tecnologia.

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239

with a possible cosmological constant, OA. Recently, this last quantity as been often substituted by QW1 where w is a constant assumed to describe the equation of state, w = p/p, of a possible dark energy component (w = — 1 in the case of a classical cosmological constant, A). We will concentrate on the cluster number density: its present-day value and evolution with redshift. We start by briefly describing why and how it can be used to constrain cosmological parameters. Next, we review the estimation of as from data on the local cluster abundance and offimbased on the redshift evolution of the cluster abundance. Finally, we discuss future prospects in this field, and show to what extent the XMM-Newton Cluster Survey (XCS) will be able to constrain cosmological parameters. 2. Cosmology with the cluster abundance The number density of dark matter haloes as a function of mass M and redshift z, also known as the halo mass function, has long been considered an essential prediction of any credible large-scale structure formation model. The (comoving) halo mass function can be written as

with M(R) — (4/3)7ri? 3 p 0 , where pm = Clmpc is the present-day total matter density (pc is the critical density), and a(R,z) = a(R,0),^f^WK(l (2) + z r , g[llm,Slw,w\ is the dispersion of the density field at some scale R at redshift z [g is a function that describes the growth of density perturbations in the linear regime, i.e. as long as <J(R,Z) < 1]). The density parameters £lm(z) and £lw(z) depend only on Qm, Qw and w. N-body simulations 1'2-3-4 have shown that if F(a) is written as F ( a ) = A e x p ( - | l n ( l / a ) + B| £ )

(3)

then the parameters A, B and e are independent of cosmology and redshift when halo masses are defined with respect to the background density. From the previous expressions one finds that n(M, z) depends essentially on (Tg, Q.m, Slw and w (in decreasing order of importance). Given that n(M, z) varies with M and z differently for distinct combinations of these parameters, in principle knowing how n(M, z) changes can lead to

240

the estimation of any of those parameters. Observationally, the most accessible interesting quantity is the number density of rich clusters of galaxies at the present epoch, providing an estimate for n(M, z ~ 0) in the range of scales 1014 to 2 x 10 15 h~x M 0 . Given that n(M, z) depends most strongly on as, traditionally the present-day abundance of galaxy clusters has been used to determine this parameter (as a function of the others). Data on the evolution with redshift of the cluster number density has in turn been used to try to break the degeneracy between
3. Current results on
flm

The best method to find clusters of galaxies is through their X-ray emission, which is much less prone to projection effects than optical identification. The selection function of X-ray cluster catalogues is therefore usually well characterized, which is essential if the cluster abundance, and hence cosmological parameters, are to be determined with small uncertainty. Further, the X-ray temperature of a galaxy cluster is at present the most reliable estimator of its mass, thereby allowing the correct comparison between the observed cluster abundance and the halo abundance predicted by structure formation models. Most determinations of as using the number density of rich clusters of galaxies at the present epoch have thus relied on X-ray selected galaxy cluster catalogues. Some recent results 5 ' 6 ' 7 are: ag = (0.7 ± 0.06) x (On/0.35)- 0 - 4 4 ; a8 = (0.66 ± 0.05) x (O m /0.35)-° 2 5 ; a8 = 0.60 + 0.17 x (Q m /0.35) - 0 - 5 ± 0.04. All results assume Clw = 1 - fim with w — —1, and are at the 95% confidence level. We have concentrated just in the case of Q m = 0.35 and found as = 0.78, being [0.72 - 1.08] the 90% confidence interval 8 . The discrepancy between the results shown is due essentially to distinct assumptions about the relation between the X-ray cluster observable, either luminosity or temperature, and the cluster mass. Note that weak lensing measurements of as show an even greater discrepancy 9 _ 1 3 . For Q.m to be be simultaneously estimated together with as, a large catalogue of galaxy clusters with measured X-ray temperatures and covering a wide range of redshifts is needed. However, such a catalogue does not exist yet, with the closest being the EMSS 14>15'16>17, RDCS 18>19 and BrightSHARC 20 - 21 . The estimates of Qm that have been obtained based on the

241

EMSScatalogue are the following: 0.55±0.17 22 ; 0.45±0.25 23 ; 0.95±0.35 24 ; 0.4-0.8 25 ; 0.25±0.10 26 ; 0.85 (> 0.35 at 2a) 27 ; 0.44±0.12 28 ; 0.2 to 0.5 29 . The analysis of the RDCS catalogue has lead to fim ~ 0.25 ± 0.15 3 0 ' 3 1 , while the analysis of the Bright-SHARC catalogue has lead to Q,m within 0.85 to 1.00 3 2 . All results have been obtained assuming fiw = 1 — ilm and w = — 1. The large dispersion in the results is to a great extent due to the limitations of the data, which exacerbates the sensitivity to differences in the various analysis.

4. The future is the XCS There is a pressing need for a new galaxy cluster catalogue, of greater size, and in particular going to higher redshift, than existing ones, and with a well understood selection function. We have described 3 3 in considerable detail how such a catalogue may be constructed through serendipitous detections of galaxy clusters in archival data from the XMM-Newton satellite. By examining the many thousands of pointings which will be made, it will be possible to build a representative sample of randomly, and hence objectively, selected X-ray clusters. The X-ray Cluster Survey (XCS) will not only be an invaluable resource for cosmological studies, but will also have a variety of other applications 3 3 . Through a likelihood analysis we were able to estimate the uncertainty associated with the estimation of cosmological parameters using those clusters of galaxies in the XCS for which we will have their X-ray temperatures reliably estimated solely from the serendipitous data 3 4 . The likelihood calculation was performed by means of a Monte Carlo method 3 5 , whereby 2000 realizations of the expected XCS catalogue were generated for an input fiducial cosmological model, with the addition of Poisson noise, and then for each realization the cosmological parameters <7g, fim and Q\, were allowed to vary so as to find their most probable values given each catalogue. The input fiducial model was the concordance cosmology with Qm = 0.3 and Cl\ = 0.7, for which we assumed a$ = 0.9. We found that the XCS will provide competitive estimates for the three most important cosmological parameters, as, O m and 17A> enabling their joint estimation to within respectively (at least) 5 per cent, 10 per cent and 15 per cent of their true values at the 95 per cent confidence. Introducing extra unknown parameters in the likelihood estimation, for example describing the equation of state of the dark energy component, would inevitably lead to some degradation in the accuracy with which all the other 3 cosmological parameters can

242 be estimated from the X C S . An accurate characterisation of t h e equation of state of the dark energy, including its possible time dependence, using the redshift evolution of the cluster abundance will require a (near) full-sky galaxy cluster survey sensitive up to a redshift of unity. Although the X C S has this sensitivity, it only covers around 1% of the sky.

References 1. A. R. Jenkins et al., MNRAS, 321, 372 (2001) 2. A. E. Evrard et a l , ApJ, 573, 7 (2002) 3. W. Hu and A. V. Kravtsov, ApJ, 584, 702 (2003) 4. E. V. Linder and A. R. Jenkins, MNRAS, 346, 573 (2003) 5. U. Seljak, MNRAS, 337, 769 (2002) 6. S. W. Allen, R. S. Schmidt and A. C. Fabian, MNRAS, 328, L37 (2001) 7. A. Voevodkin and A. Vikhlinin, ApJ, 601, 610 (2004) 8. P.T.P. Viana et a l , MNRAS, 346, 319 (2003) 9. M. L. Brown et al., MNRAS, 341, 100 (2003) 10. M. Jarvis et al., AJ, 125, 1014 (2002) 11. C. Heymans et al., MNRAS, 347, 895 (2004) 12. J. Rhodes et a l , ApJ, 605, 29 (2004) 13. L. Van Waerbeke, Y. Mellier and H. Hoekstra, A&A, 429, 75 (2004) 14. I. M. Gioia et al., ApJS, 72, 567 (1990) 15. J. P. Henry et a l , ApJ, 386, 408 (1992) 16. I. M. Gioia and G. A. Luppino, ApJS, 94, 583 (1994) 17. A. D. Lewis et al., ApJ, 566, 744 (2002) 18. P. Rosati et al., ApJ, 492, L21 (1998) 19. B. P. Holden et al., AJ, 124, 33 (2002) 20. A. K. Romer et al., ApJS, 126, 209 (2000) 21. C. Adami et a l , ApJS, 131, 391 (2000) 22. J.P. Henry, ApJ, 489, LI (1997) 23. V. R. Eke et a l , MNRAS, 298, 1145 (1998) 24. D. E. Reichart et al., ApJ, 518, 521 (1999) 25. P. T. P. Viana and A. R. Liddle, MNRAS, 303, 535 (1999) 26. M. Donahue and G. M. Voit, ApJ, 523, L137 (1999) 27. A. Blanchard et al., A&A, 362, 809 (2000) 28. J. P. Henry, ApJ, 534, 565 (2000) 29. J. P. Henry, ApJ, 609, 603 (2004) 30. S. Borgani et al., ApJ, 517, 40 (1999) 31. S. Borgani et al., ApJ, 561, 13 (2001) 32. S. C. Vauclair et al., A&A, 412, 37 (2003) 33. A. K. Romer et al., ApJ, 547, 594 (2001) 34. A. R. Liddle et al., MNRAS, 325, 875 (2001) 35. G. Holder, Z. Haiman and J. J. Mohr, ApJ, 560, L l l l (2001)

Black Hole Physics

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ACOUSTIC BLACK HOLES

VITOR CARDOSO McDonnell Center for the Space Sciences, Department of Physics, Washington University, St. Louis, Missouri 63ISO, USA and Centro de Fisica Computacional, Universidade de Coimbra, P-3004-516 Coimbra, Portugal E-mail: vcardoso @wugrav. wustl. edu We discuss some general aspects of acoustic black holes. We begin by describing the associated formalism with which acoustic black holes are established, then we show how to model arbitrary geometries by using a de Laval nozzle.

1. Introduction The progress in understanding black holes has been immense, over these last forty years since their concept was born, and they now play a central role in modern physics. An important step to make black holes more accessible (from an experimental point of view) was given in 1981 by Unruh 1, who came up with the notion of analogue black holes. While not carrying information about Einstein's equations, the analogue black holes devised by Unruh do have a very important feature that defines black holes: the existence of an event horizon. The basic idea behind these analogue acoustic black holes is very simple: consider a fluid moving with a space-dependent velocity, for example water flowing throw a variable-section tube. Suppose the water flows in the direction where the tube gets narrower. Then the fluid velocity increases downstream, and there will be a point where the fluid velocity exceeds the local sound velocity, in a certain frame. At this point, in that frame, we get the equivalent of an apparent horizon for sound waves. In fact, no (sonic) information generated downstream of this point can ever reach upstream (for the velocity of any perturbation is always directed downstream, as a simple velocity addition shows). This is the acoustic analogue of a black hole, or a dumb hole. These objects are not true black holes, because the acoustic metric satisfies the equations of fluid dynamics and not Einstein's equations. One usually expresses this by saying

245

246

that they are analogs of general relativity, because they provide an effective metric and so generate the basic kinematical background in which general relativity resides. They are not models for general relativity, because the metric is not dynamically dependent on something like Einstein's equations 2 . Following on Unruh's dumb hole proposal many different kinds of analogue black holes have been devised, based on condensed matter physics, slow light etcetera 2 . 2. Effective acoustic geometry Let us start with the equations of fluid dynamics, and try to arrange them in such a way that an effective metric stands out naturally. The fundamental equations of fluid dynamics are the equation of continuity

at + v.(pt>) = o,

(i)

and Euler's equation Pjt=p[dtv

+ (v.V)v) = -Vp + F,

(2)

where F are all the external forces acting on the fluid. Hereafter we make the following assumptions:(i) the external forces are all gradient-derived, or F = —pV$; (ii) the fluid is locally irrotational, and introduce the velocity potential ip, v = — VV>; and (iii) the fluid is barotropic, i.e., the density p is a function of pressure p only. In this case, we can define h(P)

Jo P(P') Euler's equation can be written as

P

5t^ + /i+^(VV') 2 + ^ = 0 .

(4)

To study sound waves, we follow the usual procedure and linearize the continuity and Euler's equations around some background flow, by setting p = po + tpi, p = po + epi i i/> = i/>o + ei/'i, and discarding all terms of order e2 or higher. The continuity equation yields dtpo + V.(povo) = 0 , dtp! + V.(pivo + povx) = 0. Linearizing the enthalpy we get h(po + ep\) ~ h(p0) + e # _ Inserting this in Euler's equation one gets

(5) = h0 + e^i.

-dtifo + h0 + ^(V^o) 2 + $ = 0 , px = po(dti>i + v 0 . W i ) •

(6)

247

On the other hand, since the fluid is barotropic we have pi = jfpi, and using (6) this is the same as pi = ^po(dtipi + vo-VV'i). Finally, substituting this into (5) we get 0 = -dt (-jfpo(dtipi

+ vo.VV'iH+V. (poVipi - — PoVo{dtTpi + v0.VV»i)

It can now easily be shown 2 that this equation can also be obtained from the usual curved space Klein-Gordon equation (8)

with the effective metric g^ given by 1 Poc

(9) l

2

]

-v 0 : (c 6ij -v*0v 0)_ Neither of the background quantities is assumed constant through the flow, and so they are in general dependent on the coordinates along the flow. Here we have used the definition of the local sound speed c~ 2 = | £ . We can see that the propagation of sound waves in a fluid is equivalent to the propagation of a scalar field in a generic curved spacetime described by (9). This means that all properties of wave propagation in curved space hold also for the propagation of sound waves. The power of this effective geometry should be clear: first, by changing the background flow, we change the effective acoustic metric. Second, since this geometry clearly has an apparent horizon at the point where VQ — c, and since the existence of an apparent horizon implies Hawking radiation, then there should be Hawking radiation in this geometry, which takes the form of phonons *. The Hawking temperature can be computed to yield 2 : h d(c — v±) (10) 2TT dn where v± is the component of the fluid velocity normal to the horizon, and n is the unit vector normal to the horizon. This can also be written as 1 d(c — v±) TH = l.2x l ( T 9 K m (11) dn 1000 msThis is, for all practical purposes a number too low to be detected. Despite the fact that one cannot observe Hawking radiation, one can still measure classical aspects of black holes. So we now turn to this, but first we explain how we can mimic several geometries by using a de Laval nozzle. kTH

248

3. Shaping the nozzle 3.1. The de Laval

nozzle

A de Laval nozzle is a device which can be used to accelerate a fluid up to supersonic velocities. They were first used in steam turbines, but they find many applications in rocket engines, nozzles in supersonic wind tunnels, etc. It consists of a converging pipe, where the fluid is accelerated, followed by a throat which is the narrowest part of the tube and where the flow undergoes a sonic transition, and finally a diverging pipe where the fluid continues to accelerate. It is sketched in Fig. 1.

Figure 1. A sketch of a de Laval nozzle, used to make a smooth transition from subsonic to supersonic flow. The velocity of the fluid v(x) equals the local velocity of sound c(x) at the throat of the nozzle, x = XT- The cross-section at this point is denoted by AT-

Consider now a steady, isentropic flow through the nozzle, which has a varying cross-section A(x), where x is the arc length along a streamline. Logarithmic differentiation of the continuity equation pvA = ^JR = const yields 1 dv v dx

1 dA A dx

1 dp _ pdx

(12)

For isentropic flow p = p(p) and c2 = -£-. Using this in (12) we obtain dp' Idv v dx

1 dA A dx

1 dp _ c2p dx

Using the component of Euler's equation along the streamline pv^ and combining it with Eq. (13) we have finally

(13) =

—

aj'

1 dA v2, dv (14) 2 v c dx A dx According to (14), when the flow is subsonic (v < c), dv/dx and dA/dx have opposite signs. So narrowing the pipe will make the gas flow faster, which is what we expect from common experience. In fact, for very small v,

249 Eq. (14) can be written as dv/v = -dA/A and thus vA is a constant, a well known result for incompressible fluids. The situation is opposite for v > c, when dv/dx and dA/dx have the same sign. This means that a region of increasing cross- section will accelerate the flow. 3.2. Acoustic

black holes by different

nozzle

configurations

Let us now suppose that the dependence of c on x is small, i.e., that dc/dx is not very large (as happens for example for perfect gases), and therefore that c ~ const. Then, Eq. (14) can be solved yielding I e " 2 /(2c 2 )-l/2 = A(X)/A(XT)

•

(15)

V

Let us choose the following generic form for A: A(x) = _^e[A(xT)f(x)] /2-1/2 p o r t n j s t o b e a c o n s i s t e n t solution we must have dA/dx = 0 at x = X?, which results in the constraint / ( z r ) = -£f- Eq. (15) is then trivially solved by | = f(x)A(xr) So we conclude that in order to mimic some metric, we have to be able to simulate the correct background flow. 4. A n explicit example of an acoustic black hole and classical wave phenomena in its vicinities Some classical aspects of wave propagation in acoustic black holes have been explored in 3 ' 4 . Here I will summarize some of their results, focusing always on the (2 + l)-dimensional rotating acoustic black hole 2 , which I now describe. Consider a fluid having (background) density p. Assume the fluid to be locally irrotational (vorticity free), barotropic and inviscid. From the equation of continuity, the radial component of the fluid velocity satisfies pvr ~ 1/r. Irrotationality implies that the tangential component of the velocity satisfies v8 = B/r. By conservation of angular momentum we have pv6 ~ 1/r, so that the background density of the fluid p is constant. We then take vr = A/r, and A, B are constants. The acoustic metric describing the propagation of sound waves in this "draining bathtub" fluid flow is 2 : ds2 = - (c2 -

A2

^2B2]

dt2 + ^drdt

- 2Bd<j>dt + dr2 + r2d<j>2 .

(16)

The acoustic event horizon is located at r # = A/c, and the ergosphere forms at rEs = (A2 + B2)l/2/c.

250 4.1.

QNMs

Black holes, like so many other objects, have characteristic oscillation or ringing modes, which are called quasinormal modes (QNMs) 5 , the associated frequencies being termed QN frequencies, or UJQM- The QN frequencies of the (2 + l)-rotating acoustic black hole (16), described by the effective geometry just described were recently computed in 3 ' 4 , to which we refer the reader for further details. Considering a m = 1 mode, for which the lowest mode (this is the mode that controls the ringing phase) is approximately WQN ~ (0.4 — 0 . 3 3 i ) ^ , with r # the horizon radius. If one builds an acoustic black hole by making a r # = 1 mm hole in a tub with water, then this black hole should have a characteristic ringing frequency of w ~ 4 x l 0 5 s _ 1 , and a typical damping timescale given by T = J^TJT ~ 3 X 10~ 6 s. 4.2. Late-time

tails

The evolution of a general set of initial data shows that shows that the signal from a black hole can roughly be divided in three parts: (i) the first part is the prompt response, at very early times, and the form depends strongly on the initial conditions.(ii) at intermediate times the signal is dominated by an exponentially decaying ringing phase, and its form depends entirely on the black hole characteristics, through its associated QNMs 5 . (iii) a late-time tail 6 , usually a power law falloff of the field. This power law is also highly independent of the initial data. There is another case in which wave propagation develops tails: wave propagation in odd dimensional flat spacetimes (see Cardoso et al. in 6 ) . Analogue black holes also exhibit tails, shedding their hair in a power-law falloff manner. It was found in 3 that any perturbation \I> in the vicinity of the (2 + l)-dimensional analogue black hole (16) eventually decays as v[r ~ t -(2"»+i).

(17)

On the other hand, this is precisely the tails that appear in any (2 + 1)dimensional flat spacetime 6 . We thus have a consistent and elegant result. 4.3. Superradiant

amplification

Rotating black holes can superradiate, in the sense that in a scattering experiment the scattered wave has a larger amplitude (the frequency is the same, this is not a Doppler effect as explained in Ref.7 and references therein) than the incident wave. Superradiance is a general phenomenon

251 in physics. This was discovered by Zel'dovich 7 , who pointed out that a cylinder made of absorbing material and rotating around its axis with frequency Q can amplify modes of scalar or electromagnetic radiation of frequency w, provided the condition w < mVt

(18)

(where m is the azimuthal quantum number with respect to the axis of rotation) is satisfied. The explicit numerical calculation of reflection coefficients for the (2 + l)-rotating acoustic black hole was done in Ref. 3, in the superradiant regime. 5. Conclusions Analogue black holes have proven to be a very valuable tool for the investigation of problems related to Hawking radiation. It is also possible that will yield valuable information regarding classical phenomena involving black holes. We have shown here some aspects of classical phenomena involving acoustic black holes, that may prove useful for future experimental realization of these systems. Acknowledgements I would like to take this opportunity to thank Emanuele Berti, Oscar Dias, Jose Lemos, Mario Pimenta, Ana Sousa and Shijun Yoshida for many useful conversations and collaboration. I also acknowledge financial support from FCT through grant SFRH/BPD/2004. References 1. 2. 3. 4. 5.

W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981). M. Visser, Class. Quantum Grav. 15, 1767 (1998). E. Berti, V. Cardoso and J. P. S. Lemos, Phys. Rev. D 70, 124006 (2004). V. Cardoso, J. P. S. Lemos and S. Yoshida, Phys. Rev. D 70, 124032 (2004). K. D. Kokkotas and B. G. Schmidt, Living Rev. Rel. 2, 2 (1999); H.-P. Nollert, Class. Quantum Grav. 16, R159 (1999). 6. R. H. Price, Phys. Rev. D5, 2419 (1972); E. S. C. Ching, P. T. Leung, W. M. Suen and K. Young, Phys. Rev. D52, 2118 (1995); V. Cardoso, S. Yoshida, O. J. C. Dias, J. P. S. Lemos, Phys. Rev. D 68, 061503 (2003). 7. Ya. B. Zel'dovich, Pis'ma Zh. Eksp. Teor. Fiz. 14, 270 (1971) [JETP Lett. 14, 180 (1971)]; Zh. Eksp. Teor. Fiz 62, 2076 (1972) [Sov. Phys. JETP 35, 1085 (1972)].

S U P E R R A D I A N T INSTABILITIES I N BLACK HOLE SYSTEMS*

6SCAR J. C. DIASt Perimeter Institute for Theoretical Physics, Canada. ( Also at CENTRA - Centro Multidisciplinar de Astrofisica E-mail: odias@perimeterinstitute. ca VITOR CARDOSO,

JOSE LEMOS,

)

SHIJUN YOSHIDA*

A wave impinging on a Kerr black hole can be amplified as it scatters off the hole if certain conditions are satisfied. This is known as superradiant scattering. By building a mirror around the black hole one can make the system unstable. This is the black hole bomb of Press and Teukolsky. We investigate in detail this process and compute the growing timescales and oscillation frequencies as a function of the mirror's location. It is found that in order for the system, black hole plus mirror, to become unstable there is a minimum distance at which the mirror must be located. We also show with an explicit example that such a bomb can be built, once an appropriate black hole is located. Now, a spacetime with a "mirror" naturally incorporated in it is anti-de Sitter (AdS) spacetime, since the AdS space behaves effectively as a box with a reflecting gravitational wall at its infinity. Thus, a small Kerr-AdS black hole is naturally unstable to superradiant scattering of a wave that is generated in its vicinity.

1. Superradiance. The black hole bomb The first example of superradiant scattering , which would lead to the notion of superradiant scattering in black hole spacetimes, was given by Zel'dovich l, by examining what happens when scalar waves impinge upon a rotating cylindrical absorbing object. Considering a wave of the form e-iu>t+im m c i d e n t upon such a rotating object, Zel'dovich concluded that if the frequency u of the incident wave satisfies a; < mCl, where fi is the angular velocity of the body, then the scattered wave is amplified. It was * Based on Phys. Rev. D 70, 044039 (2004), [hep-th/0404096]; and Phys. Rev. D 70, 024002 (2004), [hep-th/0405006]. t l acknowledge financial support from F C T through grant SFRH/BPD/2004. ^Respectively at: Washington University, USA; CENTRA, 1ST, Portugal; Waseda University, Japan.

252

253 also anticipated by Zel'dovich that by surrounding the rotating cylinder by a mirror one could make the system unstable. A black hole is one of the most interesting rotating objects for superradiant phenomena. If the rotating object is a Kerr black hole, then superradiant scattering also occurs 2 for frequencies w < mfl , but where fi is the angular velocity of the black hole. Feeding the amplified scattered wave again, one can extract as much energy as one likes from the black hole. Indeed, if one surrounds the black hole by a reflecting mirror, the wave will bounce back and forth, between the mirror and the black hole, amplifying itself each time. Then the total extracted energy should grow exponentially until finally the radiation pressure destroys the mirror. This is Press and Teukolsky's black hole bomb first proposed in Ref. [3], and studied in detail in Ref. [4]. Nature sometimes provides its own mirror: if one considers a massive scalar field (with mass ju) scattering off a Kerr black hole then, for (j < /x, the mass fi effectively works as a mirror 5 ; the AdS infinity also provides a natural wall 6 ; a wave propagating around rotating black branes or rotating black strings also finds itself trapped by a reflecting wall 7 . Here, following Ref. [3], we investigate in detail the black hole bomb, as proposed by Press and Teukolsky 3 , by using a scalar field model. We study the oscillation frequencies and growing timescales as a function of the mirror's location, and as a function of the black hole rotation. We also discuss, following Ref. [6], the superradiant instability of the Kerr-AdS black hole.

2. Properties of the black hole bomb A scalar wave evolving in the vicinities of a Kerr black hole (with mass M and angular momentum J = aM) is described by the curved spacetime Klein-Gordon equation V M V M $ = 0. This equation separates into an angular and radial wave equations under the ansatz $(i,r, 9, (j>) = e- iwt+im *5 / m (6»)i?(r), where Sp(0) are spheroidal angular functions, and the azimuthal number m takes on integer (positive or negative) values. For our purposes, it is sufficient to consider positive w's. Near the event horizon, r = r + , the scalar field behaves as $ ~ e -iwt e ±i(u-mn)r, ^ where the tortoise r» coordinate is defined implicitly by dr*/dr = (r 2 -I- a2)/(r2 + a2 - 2Mr), and Q = a/(2Mr+) is the angular velocity of the event horizon. Requiring ingoing waves at the horizon, which is the physically acceptable solution, one must impose a negative group velocity wgr for the wave packet. Since vgI = ± 1 we must choose the minus sign in the equation that describes the

254

scalar field near the horizon. However, notice that if w < mil, m

(1)

u

the phase velocity ~ will be positive. Thus, in this superradiance regime, waves appear as outgoing to an inertial observer at spatial infinity, and energy is in fact being extracted. Note that since we are working with positive w, superradiance will occur only for positive m, i.e., for waves that are co-rotating with the black hole. We consider a Kerr black hole bomb, a Kerr black hole surrounded by a spherical mirror (mirror placed at a constant Boyer-Lindquist radial r coordinate) with a radius ro, so that the scalar field will be required to vanish at the mirror's location, i.e., $(r = ro) = 0. With these two boundary conditions, ingoing waves at the horizon and a vanishing field at the mirror, the problem is transformed into an eigenvalue equation for u>. The frequencies satisfying both boundary conditions will be called Boxed Quasi-Normal frequencies (BQN frequencies, WBQN) a n d the associated modes will accordingly be termed Boxed Quasi-Normal Modes (BQNM). In the large wavelength limit, 1/w 3> M, for which the wavelength of the scalar wave is much larger than the typical size of the black hole, one can compute the BQNMs analytically. The procedure is based on matched asymptotic expansions. The BQN frequencies of the scalar wave that are allowed by the presence of the mirror located at r = ro can then be shown to be, in this approximation, UBQN

^

3l+l/2,n '

. .r h Id ,

,„.. (2)

r0 where ji+i/2,n are the zeros of the Bessel function Jj+i/2, and n is a natural number, labelling the mode overtone number. For example, the fundamental mode corresponds to n = 1. The imaginary component 8 is given by ,

[ J - ; - i / 2 ( j ; + l / 2 , n ) l 3l+l/2,n/ro

l4 + l / 2 ta + l/2,n)|

-

2( +1

rQ ' >

mil [

'

'

where B =

l\ Yrl(M*-a*Y 22'+! (+r„2 (21 I)--; (2f)i(2f+i)i « T i ( n ( f c

,t +4w

2A r . l2l+i ] 3i+i 2 n]

) ^

''

and 4ro2 = (ji+i/2,n - mD)2r\/{M2 - a2). The quantity J/ + 1 / 2 (j/+i/ 2 ,„) is the derivative of J; + i/ 2 (a;) evaluated at x = j;+i/ 2 ) „. Note that for large overtone n, ji+\/2,n ~ (n + i/2) 7r - Two highly important features of the black hole bomb can already be read from the equations above: first, since

255 6 oc — (Re[u>BQJv] - mfi). Therefore, 6 > 0 for Re[wBQAr] < m ^ , and 6 < 0 for Re[o>BQiv] > ?™Q. The scalar field $ has the time dependence e-iut = e-»Re(u))teit a n d t m i S ) for R e [ WBQjV ] < m n ) the amplitude of the field grows exponentially and the BQNM becomes unstable, with a growth timescale given by r = 1/8. Second, Re[wBQiv] c< 1/Vo, i.e., the wave frequency is proportional to the inverse of the mirror's radius. Thus, as one decreases the distance at which the mirror is located, the allowed wave frequency increases, and there will thus be a critical radius at which the BQN frequency no longer satisfies the superradiant condition (1). Since Re[wBQw] ~ l/j"o this means there will be a critical radius ro below which the instability is no longer present. This is just a consequence of having no superradiance. Notice also that Re[wBQJv] as given by (2) is equal to the normal mode frequencies of a spherical mirror in a flat spacetime. as expected. The differential equation for R(r) can be solved numerically to obtain the BQN frequencies. Some of our numerical results are shown in Figs. 1. Here we only show the data corresponding to the unstable BQNMs, which are the modes of interest for this discussion. ^From Fig. l.(a), where we show the imaginary part of the BQN frequency for the fundamental BQNM, one can already see two very important aspects of the black hole bomb: (i) the instability is weaker (i.e., the growing timescale I n r * r is larger) for larger mirror radius, i.e., lm[uBQN] decreases as ro increases. This is also expected on physical grounds, as was remarked by Press and Teukolsky 3 , if one views the process as one of successive amplifications and reflections on the mirror, (ii) As one decreases ro the instability gets stronger, as expected, but suddenly the BQNM is no longer unstable, i.e., the imaginary component of LJBQN drops from its maximum value to zero, and the mode becomes stable. This feature was already noticed in the discussion on the analytical results, and it changes the way we face the whole process, although it can easily be explained. The numerical results show that the analytical treatment works quite well, even though it is a large wavelength approximation. Indeed, in Fig. l.(b), we show Re{u>BQN] as a function of ro. One can see that Re[o>BQ./v] behaves as —, which is consistent with equation (2). This means that it is indeed the mirror which selects the allowed vibrating frequencies. The reason behind this minimum distance is now easily explained. The oscillation frequency is given basically by the inverse of the mirror's radius. Thus, as one decreases the distance at which the mirror is located, the oscillation frequency increases, and there will therefore be a critical radius at which the frequency no longer satisfies

256 0.001 0.0001 io- 6

x •V•v

10-" : -

io-7 10-8

a/(2U)=0.l a/(2M)=0.2 a/(2U)=0.3 a/(2U)=0.4 a/(2M)=0.4999

io-» io-'° io-" io-"

I

1

10 r,/(SM)

L

100

0.1 a/(2U)=Q.\ a/(2M)=0.2 a/(2M)-0.3 a/(2M)=0.4 a/(2M)=0.4999

!

0.01

10 r,/(2M)

100

Figure 1. (a) The imaginary part of the fundamental ( n = l ) BQN frequency, as a function of the mirror's location, ro. Here we show also the dependence on the rotation parameter. Note that at a certain critical value, this component of the BQN frequency suddenly decreases abruptly from its maximum value to zero. Therefore, the BQNM is no longer unstable. Tracking the mode to yet smaller distances shows that indeed it turns stable (i.e., the imaginary part of U>BQN gets negative). The plot refers to a I = rn = 1 wave, (b) The fundamental oscillation frequencies (real part of WBQN) as a function of the mirror's location and rotation parameter. The dots indicate at which ro the BQNM is no longer unstable [check with Fig. l.(a)]. Note that there is no perceptive o-dependence (as matter of fact there is a very small a-dependence but too small to be noticeable). Thus, the oscillation frequency basically depends only on ro, and for large ro goes as 1/ro, as predicted by the analytical formula (2). The plot refers to a I = m = 1 wave.

the superradiant condition (1). One can estimate the critical ro radius by equating the two sides, r c r j t ~ J ' ~ ^ Q ' " • This estimate for the critical radius matches very well with our numerical data. In fact, to a great accuracy r c r i t is given by the root of Re[w(r cr i t )] — mQ, = 0. Note that according to this, it is not feasible to use this process to extract energy from a very slowly rotating black hole. In fact, for very small 0, the critical radius is extremely large, thus making the task of building such a mirror an impossible one. In their description of the black hole bomb, Press and Teukolsky seem to assume that any frequency will be amplified and thus the system will become unstable no matter what frequency one deals with. For example, Press and Teukolsky consider a M = M Q and, "for ease of construction" propose to build a r 0 = 10 3 Km mirror. Our results show this bomb would explode if the rotation of the black hole satisfies a > 6 x IO - 3 , for a m = 1 mode, otherwise it is stable. We could try to go to higher m in order to

257

remedy this, because of the mfl factor. However, higher m implies higher I and our results, both numerical and analytical, show that the oscillating frequencies (Re[wsQAr]) scale with I. In fact, our numerical results show that Re[u>BQN] behaves as Re[wBQN] ~ ~n/rQ(n + 1/2). This behaviour is also predicted by the analytical study. Thus the results we have shown for m — 1 are basically the same for higher m's, apart from a scale factor. Let us give an explicit example. One must compute the typical growing timescale, since if the growing timescale is too large, this could never be used as an energy source (because it would take too long to amplify the initial energy content). Take a black hole with mass M = IAMQ and a rotation a = 0.8M. To better take advantage of the whole process, one should place the mirror at a position near the point of maximum growing rate (but larger than the point corresponding to the maximum growing rate), which in this case means ro ~ 22M ~ 45.5Km [see Fig. l-(a)]. This would give us a growing timescale of about r ~ 0.8s, i.e., every 0.8s the amplitude of the field gets approximately doubled. Notice that as energy is extracted from the black hole, its rotation a decreases, and therefore the critical radius r c r j t increases [see Fig. l.(a)]. One could worry that if the mirror is placed too close to r c r j t the mode would soon become stable and therefore energy extraction would stop. Although this can happen, it is important to note that a very small decrease in a already gives us a huge amount of extracted energy. For the example above, the energy extraction stops when the rotation decreases to a = 0.7M. As a first order calculation one may assume the process to be adiabatic and thus one can set dM ~ D,dJ, where dM and dJ are the changes in mass and angular momentum of the black hole in this process. This then gives us a total amount of ~ 0.01M of extracted energy before the bomb stops functioning, which is still a huge quantity.

3. Superradiant instability in AdS black holes A spacetime with a "mirror" naturally incorporated in it is anti-de Sitter (AdS) spacetime, which has attracted a great deal of attention recently. As is well known, anti-de Sitter (AdS) space behaves effectively as a box, i.e., the AdS infinity works as a wall. Thus, one might expect that Kerr-AdS black holes can behave as the black hole bomb just described, and that they are therefore unstable. We have shown in 6 that for small Kerr-AdS black hole this instability is indeed present (small means that the size of the black hole horizon is much smaller than the cosmological length, r+ <£.£),

258 and in fact the properties of the instabilities present in the small Kerr-AdS black hole and in the black hole-mirror system are quite similar as will be discussed in t h e next section. As shown in Ref. [8], large Kerr-AdS black holes are stable. 4. D i s c u s s i o n of t h e r e s u l t s An interesting, b u t easily understood feature born out in the work reported here 4 is t h a t there is a minimum distance at which the mirror must be located in order for the Press-Teukolsky's black hole bomb 3 to work, i.e., for the system to become unstable. Basically this is because the mirror selects the frequencies t h a t may be excited. For distances smaller t h a n this, the system is stable and the perturbation dies off exponentially. This minimum distance increases as t h e rotation p a r a m e t e r decreases. We have shown by an explicit example t h a t such a system may well be built, yielding a reliable source of energy. T h e properties of the instabilities present in t h e small Kerr-AdS black hole and in the black hole-mirror system are quite similar. T h e AdS space behaves effectively as a box, i.e., the AdS wall with typical radius I (the cosmological length) plays in t h e analogy the role of a mirror wall with radius TQ = I. In the Press-Teukolsky's black hole bomb the real p a r t of the allowed frequency is proportional to the inverse of the mirror's radius 4 , Re[w] oc 1/ro, while in the small Kerr-AdS black hole case we have found t h a t Re[w] oc l/£ 6 . Moreover, in the Press-Teukolsky's system the growth timescale of the instability satisfies 4 5~l = 1/Im[w] oc r 0 ' , while in the AdS black hole we have 6'1 oc W+U 6. References 1. Ya. B. Zel'dovich, JETP Lett. 14, 180 (1971); Sov. Phys. J E T P 35, 1085 (1972); J. D. Bekenstein and M. Schiffer, Phys. Rev. D 58, 064014 (1998). 2. J. M. Bardeen, W. H. Press and S. A. Teukolsky, Astrophys. J. 178, 347 (1972); A. A. Starobinsky, Sov. Phys. J E T P 37, 28 (1973). 3. W. H. Press and S. A. Teukolsky, Nature 238, 211 (1972). 4. V. Cardoso, O. J. G. Dias, J. P. S. Lemos and S. Yoshida, Phys. Rev. D70 044039 (2004); hep-th/0404096. 5. T. Damour, N. Deruelle and R. Ruffini, Lett. Nuovo Cimento 15, 257 (1976); S. Detweiler, Phys. Rev. D 22, 2323 (1980). 6. V. Cardoso and O. J. C. Dias, Phys. Rev. D70 024002 (2004); hep-th/0405006. 7. V. Cardoso and J. P. S. Lemos, hep-th/0412078. 8. S. W. Hawking and H. S. Reall, Phys. Rev. D 6 1 , 024014 (1999).

MICROSCOPIC BLACK HOLE D E T E C T I O N I N UHECR: THE DOUBLE BANG SIGNATURE

M.PAULOS LIP, 14-1, Av. Elias 1000-149 Lisboa

Garcia, Portugal

According to recent conjectures on the existence of large extra dimensions in our universe, black holes could be produced during the interaction of Ultra High Energy Cosmic Rays with the atmosphere. However, and so far, the proposed signatures are based on statistical effects, not allowing identification on an event by event basis, and may lead to large uncertainties. In this talk, events with a double bang topology, where the production and instantaneous decay of a microscopic black hole (first bang) is followed, at a measurable distance, by the decay of an energetic tau lepton (second bang) are proposed as an almost background free signature. The characteristics of these events and the capability of large cosmic ray experiments to detect them are discussed.

1. Introduction Recent attempts to solve the hierarchy problem rely on the existence of extra dimensions in our universe, thereby lowering the fundamental Planck scale down to TeV energies In such scenarios, gravity should get stronger at shorter distances, and therefore new phenomena should arise in TeVscale experiments. One of the most interesting of these new phenomena is the production of black holes in collisions where the centre-of-mass energy is higher than 1 TeV". In this context, Ultra High Energy Cosmic Rays (UHECR) stand out naturally as the best candidates for black hole production, through the interaction with matter in the atmosphere However, and so far, the proposed signatures of black hole production are based on statistical effects (rates and angular distributions), not allowing identification on an event by event basis, and may lead to large uncertainties In this talk events with a double bang topology are proposed as an almost background free signature, summarising the discussion done in Ref. 1, where a complete list of references can also be found.

259

260

2. The first bang: Production and decay of microscopic black holes In the proposed scenario energetic neutrinos (Ev ~ 106 - 1012 GeV) interact deeply in the atmosphere (cross-section ~ 10 3 — 107 pb) producing microscopic black holes with a mass of the order of the neutrino-parton center-of-mass energy (y/s ~ 1 - 10 TeV). The rest lifetime of these black holes is so small (T ~ 10~ 27 s) that an instantaneous thermal and democratic decay can be assumed. The average decay multiplicity (< N >) is a function of the parameters of the model (Planck mass MD, black hole mass MBH, number of extra dimension n ) and typical values of the order of 5-20 are obtained in large regions of the parameter space. A large fraction of the decay products are hadrons (~ 75%) but there is a non negligible number of charged leptons (~ 10%).The energy spectra of such leptons in the black hole centre-of-mass reference frame peaks around MBH/N. The number of extra dimensions n is constrained by existing observational data to be n > 2 We shall take n = 3 and n = 6, but the conclusions apply equally well to higher dimensions. In this study, Planck mass values MD = 1, 2 and 3 TeV were considered. The production of tau leptons in black hole decays, in particular the tau energy spectrum, was simulated using the CHARYBDIS generator, while the black holes themselves were produced according to phase-space and without assuming an explicit vN —> BH + X cross-section, as explained in Sec. 4.

3. The second bang: The decay of energetic tau leptons A detectable second bang can be produced for tau leptons with a decay length large enough for the two bangs to be well separated, but small enough for a reasonable percentage of decays to occur within the field of view. This is of course determined by the tau energy. Taus can decay leptonicaly (~ 34%) or hadronicaly (~ 66%). In both fraction of the energy escapes detection due to the presence of neutrinos. The average value of this energy, considering all decay modes, is of the order of 50%. Hadronic decays produce a large amount of visible energy, which will be seen as an extensive air shower. For leptonic decays, not only the fraction of energy not associated to neutrinos is lower, but also only decays into electrons originate extensive air showers, leading to observable fluorescence signals.

261

4. Observation p r o s p e c t s The observation window for the double bang events is constrained by geometrical considerations (the two showers must be inside the field of view) and by the signal to noise ratio. EUSO 3 will be taken as a case study, without however making any detailed simulation. Double bang events were generated parameterising the shower development and the atmosphere response, in a model implemented in MATHEMATICA 7 . This approach was inspired in the method presented in the description of the SLAST simulation 8 . The GIL parameterisation 9 was used for the longitudinal shower development, while fluorescence yield follows Ref. 10. Attenuation in the atmosphere was included using tables produced with LOWTRAN u . In Fig. 1, the longitudinal fluorescence profile of a double bang event at the entrance of EUSO is shown as an example.

Time (2.5 \isec units)

Figure 1. Longitudinal profile of double bang event, originating from an horizontal incoming neutrino with an energy of 10 2 0 eV.

A CL of 99.7% (3cr) was chosen as the criterion of visibility of the second shower. The modified frequentist likelihood ratio method,which takes into account not only the total number of expected signal and background events but also the shapes of the distributions, was used. Signal events were obtained using the method described above. Threshold energies as low as 5 x 10 18 eV (1 x 10 18 eV) can be obtained for a photon detection efficiency of 0.1 (1.0) and a shower height of 10 km.

262

The fraction of the black hole events with a first bang within the EUSO field of view that also have an observable second shower is shown in Fig. 2(a), as a function of the primary neutrino energy, for (Mrj=l TeV, n=3, MBH=5 TeV), and for detector efficiencies of 1.0 and 0.1.

20.25

20.5

20.75

21

21.25

21.5

Log,0(Ev/leV)

Figure 2. Fraction of the events with a first bang within the EUSO field of view that also have a visible second bang (a) as a function of the primary neutrino energy E„, for ( M D = 1 TeV,n = 3 , M B H = 5 TeV), and for detection efficiencies e =0.1 and e = 1 , (b) as a function of x = MBH/MD, for E = 1 0 2 0 eV and detection efficiencies e =0.1 and e = 1 . The thick lines correspond to M J J = 1 TeV,n = 3 and the dotted bands give the variation of the results when varying M u between 1 and 2 TeV and n between 3 and 6.

263

These results take into account the fraction of events with taus in black hole decays, the tau energy spectrum and its decay length, the geometrical acceptance of EUSO and the visibility of the second shower. For a detector efficiency of 0.1 (1.), at Ev = 10 20 eV, of the order of 2% (5%) of the black hole induced events with a black hole decay visible in EUSO (first bang) are expected to have a visible second bang. Figure 2(b) shows the expected fraction of double bang events as a function of x = MBH/MD, for Ev = 10 20 eV and two values of the detection efficiency, quantifying the dependence of the results on the values of the (MD,n) parameters. The rate of black hole induced events depends strongly on the assumed cosmogenic and extragalactic neutrino fluxes. Values ranging from several tens to hundreds of events per year, for Mr> = 1 TeV, have been predicted for the OWL telescope 6 . In the same reference the acceptance of EUSO was estimated to be 1/5 of OWL's. Although the expected rate of double bang events due to black hole production in EUSO is rather small and would cover a relatively small region of the parameters space, the observation of just a few such events would be of the utmost importance, as almost no standard physics process has this signature. References 1. Microscopic black hole detection in UHECR: the double bang signature", V. Cardoso et al., Astroparticle Physics Journal 22 (2005) 399-407, arXiv: hep-ph/0405056 2. Auger Collab., The Pierre Auger Project Design Report, FERMILAB-PUB96-024, 252 (1996). http://www.auger.org. 3. O. Catalano, In Nuovo Cimento, 24-C, 2, 445 (2001); http://www.eusomission.org. 4. J. F. Krizmanic et al, OWL/AirWatch Collab., Proc. of the 26*'v ICRC, Vol. 2, 388 (1999); http://owl.gsfc.nasa.gov 5. J. Tanaka, T. Yamamura, S. Asai and J. Kanzaki, Study of black holes with the ATLAS detector at the LHC, ATL-PHYS-2003-037, November 2003. 6. S. I. Dutta, M. H. Reno and I. Sarcevic, On black hole detection with the OWL/Airwatch telescope, hep-ph/0204218. 7. http://www.wolfram.com/products/mathematica/index.html 8. D. V. Naumov, SLAST: Shower Initiated Light Attenuated to the Space Telescope, LAPP-EXP-2004-02 (2002); D. Naumov, EUSO Report SDA-REP-015 (2003). 9. O. Catalano et al., Proc. of the 27"1 ICRC (2001). 10. F. Kakimoto et al., Nucl. Instrum. Meth. A372, 527 (1996). 11. F. X. Kneizys et al., The MODTRAN 2/3 report and LOWTRAN 7 Model, ed. L. W. Abreu and G. P. Anderson (1996).

GENERALIZED U N C E R T A I N T Y P R I N C I P L E A N D HOLOGRAPHY

FABIO SCARDIGLI CENTRA

- Departamento de Fisica, Institute Superior Tecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: scardigli@fisica. ist. utl.pt ROBERTO CASADIO

Dipartimento di Fisica, Universita di Bologna and I.N.F.N., Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy E-mail: [email protected]

We consider Uncertainty Principles which take into account the role of gravity and the possible existence of extra spatial dimensions. Explicit expressions for such Generalized Uncertainty Principles in 4 + n dimensions are given and their holographic properties investigated. In particular, we show that the predicted number of degrees of freedom enclosed in a given spatial volume matches the holographic counting only for one of the available generalizations and without extra dimensions.

1. Introduction During the last years many efforts have been devoted to clarifying the role played by the existence of extra spatial dimensions in the theory of gravity 1'2. One of the most interesting predictions drawn from the theory is that there should be measurable deviations from the l/r2 law of Newtonian gravity at short (and perhaps also at large) distances. Such new laws of gravity would imply modifications of those Generalized Uncertainty Principles (GUP's) (see Ref. 5) designed to account for gravitational effects in the measure of positions and energies. On the other hand, the holographic principle is claimed to apply to all of the gravitational systems. The existence of GUP's satisfying the holography in four dimensions (one of the main examples is due to Ng and Van Dam 3 ) led us to explore the holographic properties of the GUP's extended to the brane-world scenarios 4 . The results, at least for the examples we consid-

264

265

ered, are quite surprising. The expected holographic scaling indeed seems to hold only in four dimensions, and only for the Ng and van Dam's GUP. When extra spatial dimensions are admitted, the holography is destroyed. This fact allows two different interpretations: either the holographic principle is not universal and does not apply when extra dimensions are present; or, on the contrary, we take seriously the holographic claim in any number of dimensions, and our results are therefore evidence against the existence of extra dimensions. The four-dimensional Newton constant is denoted by GN throughout the paper. 2. N g and Van D a m G U P in four dimensions An interesting GUP that satisfies the holographic principle in four dimensions has been proposed by Ng and van Dam 3 , based on Wigner inequalities about distance measurements with clocks and light signals 6 . Suppose we wish to measure a distance /. Our measuring device is composed of a clock, a photon detector and a photon gun. A mirror is placed at the distance I which we want to measure and m is the mass of the system "clock + photon detector + photon gun". We call "detector" the whole system and let a be its size. Obviously, we suppose _ 2GNm a>rg = 2— = Rsirn) , (1) which means that we are not using a black hole as a clock. Be Ax\ the uncertainty in the position of the detector, then the uncertainty in the detector's velocity is Av = —^—. (2) v ; 2mAzi After the time T = 21 /c taken by light to travel along the closed path detector-mirror-detector, the uncertainty in the detector's position (i.e. the uncertainty in the actual length of the segment I) has become Aitot = Azi + T Av = A n + 7i—i—

•

(3)

We can minimize Aa; tot by suitably choosing Ax\, and we get fc/Ti

Pi rP\

/

(Aztot)min = ( A l i ) m i n + 2m(Axi)min

= 2 1 — \2mJ

I

.

(4)

Since T = 2 l/c, we have / t , x 1/2

(Axtot)min = 2 (

J

= SIQU

•

(5)

266

This is a purely quantum mechanical result obtained for the first time by Wigner in 1957 6 . Prom Eq. (5), it seems that we can reduce the error (Aa;tot)min as much as we want by choosing m very large, since (Axtot)min —> 0 for m —> oo. But, obviously, here gravity enters the game. In fact, Ng and van Dam have also considered a further source of error, a gravitational error, besides the quantum mechanical one already addressed. Suppose the clock has spherical symmetry, with a> r g . Then the error due to curvature can be computed from the Schwarzschild metric surrounding the clock. The optical path from ro > rs to a generic point r > ro is given by (see, for example, Ref. 7) ;A i

= /

TIH

/ro -"Jrn

^n

= (r ~ r°)

+ r

'

s

lo

"

r § Zr

" o -

'zT . rg

(6)

and differs from the "true" (spatial) length (r — ro). If we put a = ro, / = r, the gravitational error on the measure of (I — a) is thus I—r I Sic = rE log S- ~ r g log - , (7) a - rg a where the last estimate holds for I > a ^> r g . If we measure a distance / > 2a, then the error due to curvature is

rglog2~^p.

(8)

Thus, according to Ng and van Dam the total error is Shot = SlQM + Slc = 2 (—) * + ^ . (9) \mcj c2 This error can be minimized again by choosing a suitable value for the mass of the clock, namely mm\n = C ( ^ ) 1 , / 2 / G N and, inserting mm\n in Eq. (9), we then have («tot) m i n = 3 ( ^ 0 1 / 3 • (10) The global uncertainty on / contains therefore a term proportional to I1/3. 2.1.

Holographic

properties

We now see immediately the beauty of the Ng and van Dam GUP: it obeys the holographic scaling. In fact in a cube of size I the number of degrees of freedom is given by

n{y) =

((*du) = \WF>) = I'

as required by the holographic principle.

(11)

267

3. Models with n extra dimensions We shall now generalize the procedure outlined in a previous section to a space-time with 4 + n dimensions, where n is the number of space-like extra dimensions 4 . The link between the gravitational constant C?N in four dimensions and the one in 4 + n, henceforth denoted by G^+n), of course depends on the model of space-time with extra dimensions that we consider. Models recently appeared in the literature mostly belong to two scenarios: (I) the Arkani-Hamed-Dimopoulos-Dvali (ADD) model l , where the extra dimensions are compact and of size L; (II) the Randall-Sundrum (RS) model 2 , where the extra dimensions have an infinite extension but are warped by a non-vanishing cosmological constant. A feature shared by (the original formulations of) both scenarios is that only gravity propagates along the n extra dimensions, while Standard Model fields are confined on a four-dimensional sub-manifold usually referred to as the brane-world. In the ADD case the link between G N and G(4 +n ) can be fixed by comparing the gravitational action in four dimensions with the one in 4 + n dimensions. The space-time topology in such models is M = A44 <8> SR", where .M 4 is the usual four-dimensional space-time and K n represents the extra dimensions of finite size L. From such comparison we obtain G(4+n) ~ G N I " ,

(12)

where we omit unimportant numerical factors. The RS models are more complicated. It can be shown 2 that for n = 1 extra dimension we have G^+n) = CT~XGN, where a is the brane tension with dimensions of length*1 in suitable units. The gravitational force between two point-like masses m and M on the brane is obtained by perturbative calculations, not immediately applicable to a non-perturbative structure such as a black hole. Therefore we shall consider only the ADD scenario in this paper (see Ref. 4 for more details). 4. N g and Van D a m G U P in 4 + n dimensions Ng and van Dam's derivation can be generalized to the case with n extra dimensions. The Wigner relation (5) for the quantum mechanical error is not modified by the presence of extra dimensions and we just need to estimate the error Sic due to curvature. We ought not to consider micro black holes created by the fluctuations AE in energy, as in Ref. 5, but we have rather to deal with (more or less) macroscopic clocks and distances. This implies that we have to distinguish

268

four different cases: (1) 0 < L < rg < a < I; (2) 0 < r( 4 + n ) < L < a < I; (3) 0 < r ( 4 + „) < a < L < I; (4) 0 < r{i+n) < a < I < L; where r ( 4 + n ) is the Schwarzschild radius of the detector in 4 + n dimensions, and of course r g = T(4). The curvature error will be estimated (as before) by computing the optical path from a = ro to I = r. Of course, we will use a metric which depends on the relative size of L with respect to a and I, that is the usual four-dimensional Schwarzschild metric in the region r > L, and the 4 + n dimensional Schwarzschild solution in the region r < L (where the extra dimensions play an actual role). In cases (1) and (2) the length of the optical path from a to I can be obtained using just the four-dimensional Schwarzschild solution and the result is given by Eq. (10). In cases (3) and (4) we instead have to use the Schwarzschild solution in 4 + n dimensions n ,

+ (i - ^ r ) _1 ^ + r>dtfn+2, (is)

d* = - (i - ^ r ) ^

at least for part of the optical path. In the above, 16 7rG ( 4 + w ) m ° - ( n + 2)An+2C2 '

U4j

and An+2 is the area of the unit (n + 2)-sphere, that is A

"+* = Wn+3) •

(15)

Besides, we note that, for n = 0, 2GNm C = —j—

= rs .

( 16 )

that is, C coincides in four dimensions with the Schwarzschild radius of the detector. The 4 + n dimensional Schwarzschild horizon is located where ( l - C / r n + 1 ) = 0, that is at r =

c

i/(n+i)=r(4+n)_

(17)

Since measurements can be performed only on the brane, to the uncertainty Az in position we can still associate an energy given by fie/(2 Ax). The corresponding Schwarzschild radius is now given by Eq. (17) with m = AE/c2 and the critical length such that Ax = r-(4+n) is the Planck length in 4 + n dimensions,

Ax~(e2pLn)^=eii+n).

(is)

269 In case (3) we obtain the length of the optical path from a to I by adding the optical path from a to L and that from Ltol. We must use the solution in 4 + n dimensions for the first part, and the four-dimensional solution for the second part of the path,

°*-f( 1 + ;=£c)* + /!( 1 + ;^)*- <19> It is not difficult to show that from r( 4 + n ) < L [which holds in cases (3) and (4)] we can infer rg < r ( 4 + „) < L .

(20)

n+1

We suppose a 3> C = r?£},, that is a » T*(4+n)> so that we are not doing measures inside a black hole. Then rg < r( 4 + n ) -C a < L < I. In case (4), the optical path from a to / can be obtained by using simply the Schwarzschild solution in 4 + n dimensions. We get

cAt

= lX1 + ^^)dr = {l-a)+Cl^c-

Also here we suppose, as before, that an+1 3> C = r?^}., that is a 3> r( 4 + „) (i.e. our clock is not a black hole). If the distance we are measuring is, at least, of the size of the clock (/ > 2 a), we can minimize Sltot with respect to m in perfect analogy with the previous calculation and we obtain that the total minimum error (quantum mechanical + curvature) can be written in both cases in the form

- ^ H

.

(22)

where N(n) is an unimportant numerical factor (for detailed calculations see Ref. 4). Note that, for n = 0, Eq. (22) yields the four-dimensional error given in Eq. (10). 4.1. Holographic

properties

We finally examine the holographic properties of Eq. (22) for the GUP of Ng and van Dam type in 4 + n dimensions. Since we are just interested in the dependence of n(V) on I and the basic constants, we can write /pn+2

i\ V3

(21)

270

We then have that the number of degrees of freedom in the volume of size I is

n{V)

/

\ 3+ ™

I

//2an\1+t

=(ws-) =(w)

•

(24)

and the holographic counting holds in four-dimensions (n = 0) but is lost when n > 0. In fact we do not get something as n(V)=(-

/

I

\2+n

,

(25)

as we would expect in 4 + n dimensions. Even if we take the ideal case « ~ ^(4+n) we get / n V

()

=

i

\2(i+*)

1 » V*(4+ra)/ and the holographic principle does not hold for n > 0.

(26)

5. Concluding remarks In the previous Sections, we have shown that the holographic principle seems to be satisfied only by uncertainty relations in the version of Ng and van Dam and for n = 0. That is, only in four dimensions we are able to formulate uncertainty principles which predict the same number of degrees of freedom per spatial volume as the holographic counting. This could be evidence for questioning the existence of extra dimensions. Moreover, such an argument based on holography could also be used to support the compactification of string theory down to four dimensions, given that there seems to be no firm argument which forces the low energy limit of string theory to be four-dimensional (except for the obvious observation of our world). In this respect, we should also say that the cases (3) and (4) of Section 4 do not have a good probability to be realized in nature since, if there are extra spatial dimensions, their size must be shorter than 10 _ 1 mm 8 . Therefore, cases (1) and (2) of Section 4 are more likely to survive the test of future experiments. References 1. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998); Phys. Rev. D 59, 0806004 (1999); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 436, 257 (1998).

271 2. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999); Phys. Rev. Lett. 83, 4690 (1999). 3. Y.J. Ng and H. van Dam, Mod. Phys. Lett. A 9, 335 (1994); Mod. Phys. Lett. A 10, 2801 (1995); Phys. Lett. B 477, 429 (2000); Y.J. Ng, Phys. Rev. Lett. 86, 2946 (2001). 4. F. Scardigli and R. Casadio, Class. Quantum Grav. 20 (2003) 3915. 5. F. Scardigli, Phys. Lett. B 452, 39 (1999). 6. E.P. Wigner, Rev. Mod. Phys. 29, 255 (1957); H. Salecker and E.P. Wigner, Phys. Rev. 109, 571 (1958). 7. L.D. Landau and E.M. Lifshitz, The classical theory of fields, Pergamon Press, Oxford, 1975. 8. C D . Hoyle et al., Phys. Rev. Lett. 86, 1418 (2001). 9. S.B. Giddings, E. Katz and L. Randall, JHEP 0003, 023 (2000); J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000). 10. P.C. Argyres, S. Dimopoulos and J. March-Russell, Phys. Lett. B 441, 96 (1998). 11. R.C. Myers and M.J. Perry, Annals of Physics 172, 304 (1986).

T E S T I N G COVARIANT E N T R O P Y B O U N D S

SIJIE GAO AND JOSE P. S. LEMOS Centro Multidisciplinar de Astroffsica CENTRA Departamento de Fisica, Institute/ Superior Tecnico Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: [email protected], [email protected]

We give a brief review on Bousso's covariant entropy bound and its generalization. Bousso's bound is tested in the context of gravitational collapse. We also introduce a simple local condition for the generalized Bousso bound and apply it to a scalar field spacetime.

1. Covariant entropy bounds The original entropy bound was proposed by Bekenstein in order to rescue the generalized second law in a gedanken experiment. Consider a matter system of energy E and entropy S. Let R be the linear dimension of the system. Then the Bekenstein bound takes the form * S<2nER.

(1)

A related entropy bound conjecture is S<^,

(2)

where A is the area surrounding the system. Both the bounds above are spatial entropy bounds and can be easily violated in cosmological spacetimes 2 . The covariant entropy bound, conjectured by Bousso, is the following 3 : Consider a connected two dimensional spatial surface B with area A. Let L be a hypersurface bounded by B and generated by orthogonal null geodesies with non-positive expansion. Then the total entropy, SL, contained on L satisfies SL<J-

(3)

Evidences supporting this bound have been found in cosmological spacetimes and other matter systems 2 ~ 4 . In this conjecture, the null surface

272

273

L is required to be extended as far as possible unless a caustic is reached. This bound is tested in the context of gravitational collapse in section 2. A generalization of this bound allows L to be terminated at another spacelike 2-surface B' before a caustic is reached and the inequality (3) is replaced by5 SL<\(AB-AB>).

(4)

This is called the generalized covariant entropy bound. Some local conditions concerning this bound will be discuss in section 3. 2. Testing the Bousso bound in inhomogeneous gravitational collapse In his original paper 3 , Bousso discussed the bound in a process of gravitational collapse. In his model, a shell collapses into an existing Schwarzschild black hole. A new black hole with larger mass forms after the collapse. The apparent horizon L of the old black hole then moves in the interior of the new black hole and hits the singularity. Bousso showed that the entropy crossing L satisfies the covariant entropy bound. Deficiencies of his proof have been discussed in Ref. 6. This problem was re-examined with an exact solution of a shell filled with Tolman-Bondi dust 6 . Now we briefly review the arguments in Ref. 6. In this solution, a spherically symmetric shell with gravitational mass M collapses into a black hole with mass Mo- Due to Birkhoff's theorem, the exterior of the shell is also Schwarzschild with mass M = MQ + M. The shell consists of Tolman-Bondi dust. A marginally bound and self-similar solution for the shell is

"•'=- iT2+ *$£*%>»'* + (I s ) m (BR ~T)VW •(5) where B is a constant and the comoving radial coordinate R is restricted by i?i < R < R2. By matching the shell with its Schwarzschild interior and exterior, one may find the simple relations Ri = M0 and i?2 = M. The crucial step to test the entropy bound is to specify the entropy of the shell. We generalize the assumption in Ref. 3 where entropies for homogeneous cosmological models are defined. We assume that the entropy in a comoving volume does not change with time. In addition to this assumption, a boundary condition is needed to define the entropy. We then assume that the entropy density is equal to 1 at the hypersurface where the

274

scalar curvature is equal to 1 in Planck units. By the two assumptions, the entropy of the shell takes the simple form S = 87r(i?2 -Ri).

(6)

The entropy bound is tested numerically by comparing S with the area of the original apparent horizon A = ATT(2MQ)2. We find that the bound is satisfied for M0 > 50 and violated for M < 50. However, the apparent violation for the latter case can be interpreted as over-counting the entropy residing in the Planck regime. Therefore, there is no meaningful violation on the covariant entropy bound in classical relativity. 3. Local conditions for the generalized covariant entropy bound Flanagan et al. 5 first proved Bousso's conjecture under two independent hypotheses concerning the local entropy of matter. In the spirit of Ref. 5, alternative local sufficient conditions have been proposed 7 - 8 . In Ref. 7, it was assumed that the matter entropy can be described in terms of a local entropy current sa. Let ka be the tangent vector of the light sheet L. The expansion of ka is defined as 6 = Vaka. Let s = —kasa be the flux of entropy that crosses L and s' = kaWas be the rate of entropy flux on L. The stress-energy tensor of the matter is denoted by Tab. Then the generalized Bousso bound can be derived from the following two conditions:

(0 s' < 2TrTabkakb.

(7)

(ii) On the initial 2-surface B of the light sheet, S\B<-\9\B.

(8)

Inspired by these two conditions, we derived an even simpler local condition for the generalized entropy bound 9 . The following proposition was given in Ref. 9. Proposition 3.1. A necessary and sufficient condition for the generalized entropy bound to be satisfied for all light sheets in a region is that s < —\0 is satisfied everywhere in the region. This proposition applies to spacetimes where absolute entropy currents (entropy currents are independent of the light sheet) are well-defined. By checking s = —\Q, we shall be able to identify the regions where the bound

275

holds for all the light sheets lying inside. If the entropy density s does not vanish, proposition 3.1 indicates that there always exists a band region around the apparent horizon 9 — 0 such that the generalized bound is violated for light sheets in this region. It is worth mentioning that our proposition does not cover the case where a light sheet crosses the s = — -^9 surface. Proposition 3.1 is closely related to condition (ii) of Strominger and Thompson 7 . In Ref. 9, we showed that if both conditions in Ref. 7 are satisfied for a light sheet, then s < — \9 are satisfied everywhere on the light sheet. This implies that the role of condition (i) can be replaced by the single condition s < — ^9. It can be shown that there exist light sheets on which s < — \9 is satisfied but condition (i) is nowhere satisfied. Therefore, condition (i) is not a necessary condition for the generalized entropy bound. To demonstrate how Proposition 3.1 works, we consider a scalar field spacetime which was first discussed by Husain 4 . The spacetime is described by the metric ds2 = -tf{r)

dt2 +tf{r)-1

dr2 +tr2 / ( r ) _ 1 W ^ [d92 + sin 2 9dcj>2),

where -—

1

m = { -l)

2

-

(9)

and the corresponding scalar field in the spacetime is


(10)

4

sa = -dacj).

(11)

Using this expression, we test the condition s < — \9 for future-directed ingoing light sheets. The result is shown in Fig.l. We see that there exists a band region between the apparent horizon and the s = — ^9 surface where s > — \9. Proposition 3.1 then indicates that the generalized entropy bound is violated for any light sheet residing in this region. However, such kind of violation could be caused by the failure of hydrodynamic description of matter entropy 8 . To justify the violation, one needs to compare the proper length of the light sheet with the thermal wavelength of the scalar field. Detailed discussion can be found in Ref. 9.

276

10

15

20

25

30

Figure 1. A spacetime diagram depicting the regions where s < — \0 is satisfied and violated. The generalized entropy bound is satisfied in the region s < — \6 and violated in the region s > — \6.

Acknowledgments This work was partially funded by Fundacao para a Ciencia e Tecnologia (FCT) - Portugal through project POCTI/FNU/44648/2002. SG acknowledges financial support by the FCT grant SFRH/BPD/10078/2002 from FCT. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

J. D. Bekenstein, Phys. Rev. D 9, 3292 (1974). R. Bousso, Rev. Mod. Phys. 74, 825 (2004). R. Bousso, J. High Energy Phys. 9907, 004 (1999). V. Husain, Phys. Rev. D 69, 084002 (2004). E. E. Flanagan, D. Marolf, and R. M. Wald, Phys. Rev. D 62, 084035 (2000). S. Gao and J. P. S. Lemos, J. High Energy Phys. 0404, 017 (2004). A. Strominger and D. Thompson, Phys. Rev. D 70, 044007 (2004). R. Bousso, E. E. Flanagan and D. Marolf, Phys. Rev. D 68, 064001 (2003). S. Gao and J. P. S. Lemos, Phys. Rev. D, accepted for publication, (2005).

Dark Matter and Dark Energy

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D A R K E N E R G Y - D A R K M A T T E R UNIFICATION: GENERALIZED CHAPLYGIN GAS MODEL

ORFEU BERTOLAMI Instituto

Superior Tecnico, Departamento de Av. Rovisco Pais, 1049-001 Lisboa, Portugal E-mail: [email protected]

Fisica

We review the main features of the generalized Chaplygin gas (GCG) proposal for unification of dark energy and dark matter and discuss how it admits an unique decomposition into dark energy and dark matter components once phantom-like dark energy is excluded. In the context of this approach we consider structure formation and show that unphysical oscillations or blow-up in the matter power spectrum are not present. Moreover, we demonstrate that the dominance of dark energy occurs about the time when energy density fluctuations start evolving away from the linear regime.

1. Introduction The GCG model 1,2 is an interesting alternative to more conventional approaches for explaining the observed accelerated expansion of the Universe such as a cosmological constant 3 or quintessence4. It is worth remarking that quintessence is related to the idea that the cosmological term could evolve5 and with attempts to tackle the cosmological constant problem. In the GCG approach one considers an exotic equation of state to describe the background fluid: A Pch = —-sr ,

(1)

Pch

where A and a are positive constants. The case a — 1 corresponds to the Chaplygin gas. In most phenomenological studies the range 0 < a < 1 is considered. Within the framework of Friedmann-Robertson-Walker cosmology, this equation of state leads, after being inserted into the relativistic energy conservation equation, to an evolution of the energy density as 2 B

279

1T+"

280

where a is the scale-factor of the Universe and B a positive integration constant. From this result, one can understand a striking property of the GCG: at early times the energy density behaves as matter while at late times it behaves like a cosmological constant. This dual role is what essentially allows for the interpretation of the GCG model as an entangled mixture of dark matter and dark energy. The GCG model has been successfully confronted with different classes of phenomenological tests: high precision Cosmic Microwave Background Radiation data 6 , supernova data 7 , and gravitational lensing8. More recently, it has been shown using the latest supernova data 9 , that the GCG model is degenerate with a dark energy model with a phantom-like equation of state 1 0 , 1 1 . Furthermore, it can be shown that this does not require invoking the unphysical condition of violating the dominant energy condition and does not lead to the big rip singularity in future 10 . It is a feature of GCG model, that it can mimic a phantom-like equation of state, but without any kind of pathologies as asymptotically the GCG approaches to a well-behaved de-Sitter universe. Structure formation has been studied in Refs. [2, 12]. In Ref. [13], the results of the various phenomenological tests on the GCG model are summarized. Despite these pleasing performance concerns about such an unified model were raised in the context of structure formation. Indeed, it has been pointed out that one should expect unphysical oscillations or even an exponential blow-up in the matter power spectrum at present 14 . This difficult arises from the behaviour of the sound velocity through the GCG. Although, at early times, the GCG behaves like dark matter and its sound velocity is vanishingly small as one approaches the present, the GCG starts behaving like dark energy with a substantial negative pressure yielding a large sound velocity which, in turn, produces oscillations or blow-up in the power spectrum. In any unified approach this is inevitable unless the dark matter and the dark energy components of the fluid can be properly identified. These components are, of course, interacting as both are entangled within a single fluid. However, it can be shown that the GCG is a unique mixture of interacting dark matter and a cosmological constant-like dark energy, once one excludes the possibility of phantom-type dark energy 15 . It can be shown that due to the interaction between the components, there is a flow of energy from dark matter to dark energy. This energy transfer is vanishingly small until recent past, resulting in a negligible contribution at the time of gravitational collapse (zc ~ 10). This feature makes the model indistinguishable from a CDM dominated Universe till recent past. Subse-

281 quently, just before present (z ~ 2), the interaction starts to grow yielding a large energy transfer from dark matter to dark energy, which leads to the dominance of the latter at present. Moreover, it is shown that the epoch of dark energy dominance occurs when dark matter perturbations start deviating from its linear behaviour and that the Newtonian equations for small scale perturbations for dark matter do not involve any fc-dependent term. Thus, neither oscillations nor blow-up in the power spectrum do develop. 2. Decomposition of the GCG fluid In Ref. [2], it is shown that the GCG can be described through a complex scalar field whose Lagrangian density can be written as a generalized BornInfeld theory: LGBI

=

~A^

GTMW*1

,

(3)

which reduces into the Born-Infeld Lagrangian density for a = 1. The field 0 corresponds to the phase of the complex scalar field2. Let us now consider the decomposition of the GCG into components. Introducing the redshift dependence and using Eqs. (1) and (2), the pressure is given by Pch =

^ :

(4)

[4 + £(1 + «) 3 < 1+a >] 1+Q

while the total energy density can be written as l

Pch =

A + B(l + z)3<1+a>

1+a

,

(5)

where the present value of the scale-factor, ao, has been set to 1. We decompose the energy density into a pressure-less dark matter component, p4m, and a dark energy component, px, with an equation of state px = wxPx', hence the equation of state parameter of the GCG can be written as Pch Px wxpx ,cs w = — = = . (6) Pch Pdm + PX Pdm + PX Therefore, from Eqs. (4), (5) and (6), one obtains for px Pdm PX

,-s

~ l + W x [ l + f ( l + z )3(l+a)] • V) ^From this equation one can see that requiring that px > 0 leads to the constraint wx < 0 for early times ( z > l ) and wx < — 1 for the future

282

(z = —1). Thus, one can conclude that wx < —1 for the entire history of the Universe. The case wx < - 1 corresponds to the so-called phantom-like dark energy, which violates the dominant-energy condition and leads to an ill defined sound velocity (see however Ref. [10]). Excluding this possibility, then the energy density can be uniquely split as

(8)

P = Pdm + PA

where fl(l + z) 3 ( 1+a > Pdm =

(9)

[A + B(l + z)3(l+a)] 1+5 and PA = -PA =

=5T

.

(10)

[A + B(l + z) 3 (!+ a )] i+o from which one finds the scaling behaviour of the energy densities ^i PA

=

^(l A"

+ 2 )3(l+a)

_

'

( n ) K

1 r-T—i—i—t—i—i—i—i—i—i—i—i—i—i—i—i—r-

0.8

— n. —. n'dm

Figure 1. Density parameters Qdm a n d CI A and fij, as a function of redshift. It is assumed that H^o = 0.05, n d m Q = 0.25, HAQ = 0.7 and a = 0.2.

283

In what follows we express parameters A and B in terms of cosmological observables. Prom Eqs. (9) and (10), it implies that (12)

PchO = PdmO + PAO = {A + B) 1+°

where pcho, PdmO and p\o are the present values of pch, pm and pA, respectively. Constants A and B can then be written as a function of pcho A = PAO PchO !

B

(13)

= PdmO PelchO

It is also interesting to express A and B in terms of Qdmo, ^AO, the present values of the fractional energy densities fidm(A) = Pm(A)/Pc where pc is the critical energy density, pc = 3H2/8irG. From the Friedmann equation 3H2 = 8TTG \A + B(l + z) 3 < 1 + a ) ]

1+Q

+ SnGpb0{l +

zf

(14)

where pbo is the baryon energy density at present, one obtains A ~ f i A 0 ^ + a ) , B ~ Qdm0 Pc\+a)

•

(15)

Therefore, with the present value of the Hubble parameter, HQ-

nA0 + ndmo(i + *) 3(1+Q)

H2 = H2

1 +a l+a

+ n M (i + z)

(16)

one can write the fractional energy densities Vldm, ^A and Clb as

n,dm nA =

* W i + *) 3(1+Q)

(17)

3il+a) a/(1+a)x

[nA0 + ndm0(i +

z)

}

^A0

z)^+^}a/il+a)x

[nAO + iidmo(i + n6 =

nb0{i + zf

(18)

(19)

x

where X

fiAO + ^m0(l +

Z)^1+^

!/(!+«)

+ nbQ(i + z)3

(20)

Finally, as QdmO a n d ttAo are order one quantities, one can easily see that at the time of nucleosynthesis, QA is negligibly small, and hence the model is not in conflict with known processes at nucleosynthesis.

284

It is important to realize that there is an explicit interaction between dark matter and dark energy. This can be understood from the energy conservation equation, which in terms of the components can be written as Pdm + 3Hpdm = - p A .

(21)

Thus, the evolution of dark energy and dark matter are coupled so that energy is exchanged between these components (see Refs. [16,17] for earlier work on the interaction between dark matter and dark energy). One can see from Figure 1, that until z ~ 2, there is essentially no exchange of energy and the A term is vanishingly small. However, around z ~ 2, the interaction starts to increase, resulting in a substantial growth of the dark energy term at the expense of the dark matter energy. Thus, by around z ~ 0.2, dark energy starts dominating the energy content of Universe. Of course, these redshift values are a dependent and, in Figure 1, a = 0.2 has been chosen. Nevertheless, the main conclusion is that in this unified model, the interaction between dark matter and dark energy is vanishing small for almost the entire history of the Universe making it indistinguishable from the CDM model. As can be clearly seen, the energy transfer has started in the recent past resulting in a significant energy transfer from dark matter to the A-like dark energy. In the next section we show that this energy transfer epoch is the one when dark matter perturbations start departing from its linear behaviour. But before that notice also that Eq. (21) expresses the energy conservation for the background fluid, which is reminiscent of earlier work on varying A cosmology 5,18 ' 19 where the cosmological term decays into matter particles. In here, we have the opposite, as a is always positive, hence the energy transfer is from dark matter to dark energy. This is responsible for the late time dominance of the latter and ultimately to the observed accelerated expansion of the Universe. 3.

Structure formation

Aiming to study structure formation, it is interesting to write the 0-0 component of Einstein's equation as 3H2 = 8itG(pdm + pb) + A ,

(22)

where A is given by A = SnGpA

.

(23)

We address now the issue of energy density perturbations. We first write the Newtonian equations for a pressure-less fluid with background density

285 1

10

- i i 11111

-|—T T T I III]

1

1

i.

r // // •'

/

/

/

.

.••

/ / . - • • " "

1r

0.1

-

r

0.01

\ • i 11 n l

0.001

0.01

i

1 0.1

'

Figure 2. Density profile S^m. as function of scale factor. The solid, dotted, dashed and dash-dot lines correspond to a = 0,0.2,0.4,0.6, respectively. It is assumed that n 6 0 = 0.05, n d m 0 = 0.25 and fiAo = 0.7.

Pdm and density contrast Sdm, with a source term due to the energy transfer from dark matter to dark energy. Assuming that both, the density contrast Sdm and the peculiar velocity v are small, that is 5dm < < 1 and v « u, where u is the velocity of a fluid element, one can write the Euler, the continuity and the Poisson's equations in the co-moving frame 19 : dv a V$ ax+ — + -v = at a a d5dm Vddn V-w dt Pdn 1

V 2 $ = ±irGpdm{l + Sdm) - A

(24) (25) (26)

where $ is the gravitational potential, and * is the source term in the continuity equation due to the energy transfer between dark matter and the cosmological constant-type dark energy. The co-moving coordinate x is related to the proper coordinate r by r = ax. In here,

One expects a perturbation also in the A term. However, it can be seen from the Euler equation, for a fluid with an equation state of the form

286 p = wp, dv (w + \)p [ — + v • Vv ) + wVp +(w + l ) p V $ = 0

(28)

so that, for w = — 1, it follows that Vp = 0, from which implies that this cosmological constant like component is always homogeneous. We should mention that the Euler Eqs. (26) and (28) can have an extra term in the r.h.s. if the velocity of the created A-like particle has a different velocity from the decaying dark matter particle 19 . In this case, the A-like dark energy can have spatial variations which can be neglected for the Newtonian treatment. However, in our case, we are considering only the situation where both the decaying and created particles have the same velocity. i,From the divergence of Eq. (24) and using Eqs. (25) and (26), one obtains the small scale linear perturbation equation for the dark matter in the Newtonian limit: d26dn •+ dt2

2- + a

Pdm

dt

AnGpdm - 2 a Pdm

* — Pdn dt

6dm = 0 .(29)

One sees that, if \P = 0, that is in the absence of energy transfer, one recovers the standard equation for the dark matter perturbation in the ACDM case. One can verify that this occurs for a = 0. It can also be seen from the above equation that there is no scale dependent term to drive oscillations or to cause any blow up in the power spectrum. We turn now to the evolution for the baryon perturbations in the Newtonian limit when the scales are inside the horizon. Given that our purpose is to consider the period after decoupling, the baryons are no longer coupled to photons and one can effectively consider baryons as a pressure-less fluid like the dark matter as there is no significant pressure due to Thompson scattering. We assume that there is no interaction between dark energy and baryons, which means that the Equivalence Principle is violated as, on its turn, dark matter and dark matter are strongly coupled. Given that it is parameter a that controls this interaction (a = 0 means there is no interaction), it is a measure of the violation of the Equivalence Principle. One can also see from the behaviour of \&, that this violation also starts rather late in the history of the Universe. In the Newtonian limit, the evolution of the baryon perturbation after decoupling for scales well inside the horizon is similar to the one for dark matter however, as described earlier, the source term is absent as there is no energy transfer to or from baryons.

287 2 1.8 1.6 1.4

1

r

1 1 —1 1 1 1 1

1

1

r 1 1 1 11

1

1 1

" " : . / /—

~— ~ 1_ -0.8 . 0.6

1 M 1

: : _ -:

/ ,*-

*'

••..,-

'*

.

\ ~ \ -

~

\

•

t

"6T001

. , , ,,,l

0.01

,

.

. ....1

"

0.1

.

, .

•

1

1

1

1

a Figure 3. The growth factor m(y) as a function of scale factor a. The solid, dotted, dashed and dash-dot lines correspond to a = 0,0.2,0.4,0.6, respectively. It is assumed that Ub0 = 0.05, Qdmo = 0.25 and n A 0 = 0.7.

Thus, the equation for the evolution of baryon perturbations is given by d2Sb , nadSb . ^ 5 - + 2 — — - AnGpdmSdm = 0 , (30) at1 a at where in the third term in the l.h.s., the contribution from baryons has been dropped as it is negligible compared to the one of dark matter. It is convenient to define for each component the linear growth function D(y), 6 = D(y)50

,

(31)

where y = log(o) and 6Q is the initial density contrast (assuming a Gaussian distribution). It is also interesting to consider the so-called growth exponent m(y) = D (y)/D(y), where the prime denotes derivative with respect to the scale factor. Asymptotically, given that dark matter drives the evolution of the baryon perturbations, then they grow with the same exponent m(y). However, their amplitudes may differ and their ratio corresponds to the so-called bias parameter, b = Sb/SdmIt is of course, phenomenologically interesting to study the behaviour of 5dm, rn(y) and b as function of the scale factor a. While solving the

288 1.11

1

1—i i I T I I |

i—i—i r i i i q

1—i—i i m i

\-.

\\ 0.9 -

Vr i * '• •A :

1

-

';> -

*

i 0.8 -

'\\ ii

\ 0.7 -

n 61

0.001

i

i

I

1—i—•

0.01

• • ' • ' !

0.1

1—i—i

i i i 11

1

a Figure 4. Bias parameter b as a function of the scale factor, o. The solid, dotted, dashed and dash-dot lines correspond to a = 0,0.2,0.4,0.6, respectively. It is assumed that Qb0 = 0.05, QdmO - 0.25 and fiAo = 0.7.

differential equations for the linear perturbation, the initial conditions are chosen so that at a = 1 0 - 3 , the standard linear solution D ~ a is reached. In Figure 2, it is shown the linear density perturbation for dark matter, 5dm, as a function of a. One sees that, whereas for a = 0 (the ACDM case), the perturbation stops growing at late times, for models with a > 0 the perturbation starts departing from the linear behaviour around z ~ 0.25, the very epoch when the A term starts dominating (cf. Figure 1). In view of this behaviour, it is tempting to conjecture that, in our unified model, the interaction between dark matter and A-like dark energy is related with structure formation, so that for a sufficiently high density contrast (Sdm » 1), a significant energy transfer from dark matter to dark energy takes place. In any case, our proposal for GCG indicates that there is a connection between structure formation scenario and the dominance of dark energy, a link that ultimately results in the acceleration of the Universe expansion. This feature hints a possible way to understand why O ^ ~ Q\ just at recent past, the so-called Cosmic Coincidence problem. The behaviour of m(y) is also interesting. One can infer from Figure 3 that from ^ ~ 5 to the present, the growth factor is quite sensitive to the value of a. For a = 0.2, m(y) increases up to 40% at present in relation to

289 0.4

0.38

0.36 0.34 0.32 aE 0.3 0.28 0.26 0.24 0.22 0.2 0

0.05

0.1

0.15 a

0.2

0.25

0.3

Figure 5. Contours for parameters 6 and m in the Ctm-a plane. Solid lines refer to b whereas dashed lines refer to m. For b, contour values are 0.98, 0.96, ..., 0.9 from left to right. For m, contour values are 0.6, 0.65, ..., 0.8 from left to right.

the ACDM case. Notice that m(y) governs the growth of the velocity fluctuations in the linear perturbation theory as the velocity divergence evolves as —HamSdm', it follows then that large deviations of the growth factor with changing a are detectable via precision measurements of large scale structure and associated measurements of the redshift-space power spectrum anisotropy. In what concerns the bias parameter, its behaviour is shown in Figure 4. From there one can see that it also changes sharply in the recent past as a increases. This bias extends to all scales consistent with the Newtonian limit, hence being distinguishable from the hydrodynamical or nonlinear bias which takes place only for collapsed objects. Therefore, from the observation of large scale clustering one can distinguish the non-vanishing a case from the a = 0 (ACDM) case. The growth factor and the bias parameter at z ~ 0.15 have been recently determined using the 2DF survey 20,21 . It is found for the redshift space distortion parameter, f3 = 0.49 ± 0.09, and for the linear bias, b — 1.04 ± 0.14. Notice that, as j3 = m/b, one can obtain m = 0.51 ± 0.11. In Figure 5, it is shown contours for b and m in the Q m -a plane. ^From the mentioned observational constraints on b and m, one can constrain a to a small but non-zero value (a ~ 0.1). However, it is important to point out that our study refers to the properties of the baryons whereas the observations concern the fraction of baryons that collapsed to form bright

290 galaxies; the relation between the two is still poorly known. As far as parameter (3 is concerned, one should bear in mind that this constraint is obtained in the context of the standard ACDM model in order to convert redshift to distance. Thus, a full analysis in the context of the GCG model is still to be performed. Furthermore, as can be seen from Figure 2, there is no suppression of 8dm at late times for any positive value of a, and hence one should not expect the corresponding suppression in the power spectrum normalization,
4. Conclusions In this contribution, we have presented a setup where the GCG has been decomposed in two interacting components. The first one behaves as dark matter since it is pressure-less. The second one has an equation of state, Px = uxPx- It has been shown that UJX < — 1- Thus, once phantom-like behaviour is excluded the decomposition is unique. Apparently the model does not look different from the interacting quintessence models where one has two different interacting fluids; however, an interesting feature of our proposal is that it can be described through a single fluid equation. Hence, as far the background cosmology is concerned, we have an unified GCG fluid behaving as dark matter in the past and as a dark energy in the present. Nevertheless, when studying structure formation in this model one should consider it as an interacting mixture of two fluids to achieve a proper description. In any unified model, one expects an entangled mixture of interacting dark matter and dark energy. In the case of the GCG, we can uniquely identify the components of this mixture and the interaction. Moreover, we find that one does not need anything besides an evolving cosmological term to describe dark energy. This is consistent with recent studies that show that a combination of WMAP data and observations of

291 high redshift supernovae can be described via a cosmological constant-like dark energy 23 . One can also consider the GCG as a decaying dark matter model where the decay product is a cosmological constant. Obviously it remains to be seen how one can obtain such a decaying dark matter model from a fundamental theory. Given the fact that the GCG equation of state arises from a generalized Born-Infeld action, it is possible that D-brane physics can shed some light into this issue (see eg. Ref. [24]). Furthermore, we have demonstrated that in the context of our setup, the so-called dark energy dominance is related with the time when matter fluctuations become large (<5<jm > 1), a possibility has actually been previously conjectured 25 . Moreover, we have shown that in what concerns structure formation, the linear regime {5dm ~ o) is valid till fairly close to the present, meaning that at the time structure formation begins, zc ~ 10, the influence of the dark energy component was negligible and that clustering occurs very much like in the CDM model. We have shown that the growth factor as well as the bias parameter have a noticeable dependence on the a parameter. We have implemented a model which exhibits a violation of the Equivalence Principle, as dark energy and baryons are not directly coupled. This may turn out to be an important observational signature of our approach. Acknowledgments It is a pleasure to thank Maria Bento, Anjan Sen, Somasri Sen and Pedro Silva for sharing the fun on the research of the GCG properties. References 1. A. Kamenshchik, U. Moschella and V. Pasquier, Phys. Lett. 511 (2001) 265. 2. M.C. Bento, O. Bertolami and A.A. Sen, Phys. Rev. D66 (2002) 043507. 3. See e.g. M.C. Bento and O. Bertolami, Gen. Relat. and Gravitation 31 (1999) 1461; M.C. Bento, O. Bertolami and P.T. Silva, Phys. Lett. B498 (2001) 62. 4. B. Ratra and P.J.E. Peebles, Phys. Rev. D37 (1988) 3406; Ap. J. Lett. 325 (1988) 117; C. Wetterich, Nucl. Phys. B302 (1988) 668; R.R. Caldwell, R. Dave and P.J. Steinhardt, Phys. Rev. Lett. 80 (1998) 1582; P.G. Ferreira and M. Joyce, Phys. Rev. D58 (1998) 023503; I. Zlatev, L. Wang and P.J. Steinhardt, Phys. Rev. Lett. 82 (1999) 986; P. Binetruy, Phys. Rev. D60 (1999) 063502; J.E. Kim, JHEP 9905 (1999) 022; J.P. Uzan, Phys. Rev. D59 (1999) 123510; T. Chiba, Phys. Rev. D60 (1999) 083508; L. Amendola, Phys. Rev.. D60 (1999) 043501; O. Bertolami and P.J. Martins, Phys. Rev. D61 (2000) 064007; Class. Quantum Gravity 18 (2001) 593; A.A. Sen, S. Sen

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7.

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10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

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COSMOLOGY A N D SPACETIME SYMMETRIES

RALF LEHNERT Department of Physics and Astronomy Vanderbilt University, Nashville, Tennessee, 37235 E-mail: ralf. lehnert@vanderbilt. edu

Cosmological models often contain scalar fields, which can acquire global nonzero expectation values that change with the comoving time. Among possible consequences of these scalar-field backgrounds, an accelerated cosmological expansion, varying couplings, and spacetime-symmetry violations have recently received considerable attention. This talk studies the interplay of these three key signatures of cosmologically varying scalars within a supergravity framework.

1. I n t r o d u c t i o n Sizable efforts are currently directed towards the search for a more fundamental theory that must resolve a variety of theoretical issues left unaddressed in present-day physics. Many approaches along these lines require novel scalars as a key ingredient. Moreover, scalar fields are often invoked in cosmological models 1 , 2 ' 3 to explain certain phenomenological questions, such as the observed late-time accelerated expansion of the Universe, the horizon and the flatness problem, or the claimed variation of the fine-structure parameter. The search for observational consequences of such scalars can therefore yield valuable insight into new physics. Scalar fields can acquire global expectation values that vary with the expansion of the Universe. Measurable effects of variations in the scalar background are typically feeble because of the cosmological time scales involved. This suggests ultrahigh-precision studies as a promising tool for experimental investigations. The remaining task is to identify suitable tests. Varying scalars determine a spacetime-dependent background that violates translation invariance. This effect could be measurable because symmetries are typically amenable to high-precision tests. Moreover, translation-invariance breakdown is typically associated with Lorentz violation 4 , 5 since translations, rotations, and boosts are intertwined in the Poincare group. This perhaps less appreciated result opens a door for al-

293

294 ternative experimental investigations of cosmologically varying scalars and is the primary focus of this talk. Lorentz and CPT breakdown has also been suggested in other contexts as a candidate Planck-scale signature including strings,6 spacetime foam,7'8 nontrivial spacetime topology,9 loop gravity,10 and noncommutative geometry.11 The emergent low-energy effects are described by the Standard-Model Extension (SME),12 which has provided the basis for studies of Lorentz and CPT breaking with mesons, 13,14,15,16 baryons, 17,18,19 electrons, 20,21,22 photons, 23 muons,24 neutrinos, 12,25 and the Higgs.26 2. Connection between translation and Lorentz symmetry Consider the angular-momentum tensor J1*" = J d3x {d0lJ,xv — O^x11), which generates boosts and rotations. Its construction involves the energymomentum tensor 9^", which is no longer conserved when spacetimetranslation symmetry is broken. Consequently, J^u will typically depend on time, so that the usual time-independent boost and rotation generators cease to exist. As a result, Lorentz and CPT invariance is no longer assured. Lorentz violation through varying scalars can also be understood in a more intuitive way. A varying scalar is always associated with a nonzero 4-gradient. This background gradient determines a preferred direction on scales comparable to the variation: Consider, for instance, a particle species that is coupled to such a gradient. The propagation features of these particles might now depend upon whether the motion is perpendicular or parallel to the background gradient. This implies physically inequivalent directions, and thus the violation of rotation invariance. Since rotations are contained in the Lorentz group, Lorentz symmetry must be broken. We finally establish the effect at the Lagrangian level. Consider a model with a varying coupling £(x) and two scalars cj> and $. Suppose that the Lagrangian contains a term £(i)9M(/>9M$. We next integrate the action by parts with respect to the partial derivative acting on <j>. This produces an equivalent Lagrangian CJ D —if <9M$. Here i f = d^£ is a nondynamical background 4-vector violating Lorentz invariance. Note that for cosmological variations of £ we have i f = const, on small scales. 3. Toy supergravity cosmology We now illustrate the result from Sec. 2 with a toy model. This model is motivated by pure N = 4 supergravity in four spacetime dimensions. Although unrealistic in detail, it is a limit of N = 1 supergravity in eleven

295 dimensions, which is contained in M-theory. One thus expects that our model can illuminate generic aspects of a candidate fundamental theory. Suppose that only one of the model's graviphotons, F^", is excited. Then, the bosonic part of pure N = 4 supergravity is given by 2 7 ' 4 K£sg

= -\y/gR

+ ^(d^Ad^A

-\Ky/gMFliVF'iV

+

d^Bd»B)/4B2

- lity/gNF^F^

,

(1)

where M and N are known functions of the scalars A and B. The dual field-strength tensor is F^ = EIJ-upaFpa/2 and g = -det(g^), as usual. Note that we can rescale F^v —> F^" j \[K removing the explicit appearance of the gravitational coupling K from the equations of motion. For phenomenological reasons, we also include 5C = — \^fg{m2AA2 + 2 m BB2) into £ S g- 5 We further represent the model's fermions 27 by the energy—momentum tensor of dust T^,v = pu^Uv describing, e.g., galaxies. Here, uM is a unit timelike vector and p is the fermionic energy density. The usual assumption of an isotropic homogeneous flat FriedmannRobertson-Walker Universe implies that F^v = 0 on large scales. Our cosmology is then governed by the Einstein equations and the equations of motion for the scalars A and B. At tree level, the fermionic matter is not coupled to the scalars, so we can take T^v as covariantly conserved separately. Searching for solutions with this input yields a nontrivial dependence of A and B on the comoving time t.

4. Lorentz v i o l a t i o n Consider small localized excitations of F^v in the scalar background Ab and Bb from Sec. 3. Since experiments are usually confined to a small spacetime region, it is appropriate to work in local inertial coordinates. The effective Lagrangian £ c o s m for such situations follows from Eq. (1) and is £cOSm = -\MbFllvFi"'

- \NhF^F^

,

(2)

where Ab and Bb imply the time dependence of Mb and Nb. Comparison with the usual Maxwell Lagrangian £ e m = -j^F^F^ j^F^F^" 2 2 shows that e = 1/M b and 9 = An Nh. Thus, e and 6 acquire time dependencies, as they are determined by the varying background Ab and Bb, The breakdown of Lorentz symmetry in our effective Lagrangian (2) is best exhibited by the resulting modified Maxwell equations:

^F^

~ J r ^ ) ^ + ^O^F^

=0 .

(3)

296 In our toy cosmology, the gradients of e and 9 are nonzero, approximately constant locally, and act as a nondynamical external background. Thus, the gradients select a preferred direction in local inertial frames violating Lorentz invariance. The term containing d^O can be identified with the k,AF operator in the minimal SME. This term has recently received a lot attention. 2 8 For instance, it can lead to vacuum Cerenkov radiation. 2 9 Next, we consider small localized excitations 5A and SB of the scalar background in local inertial coordinates at the point XQ. With the ansatz A{x) = A\>{x) + 5A(x) and B{x) = B\,{x) + SB(x) in the equations of motion for A and B, we find the linearized equations 0 = [ • - 2 5 ^ + 2m2ABl}5A 0 = [2Ai*dr]6A +[D-

- [2A»d^ - 2A>iBli - 4m2AAbBh]SB

2B»dti + 6 m | B g - A M M + B^B^SB

,

, (4)

where Ah, Bh, A^ = B^d^A^, and B* = B^d^B^ are evaluated at x0. Equation (4) governs the propagation of 5A and 5B in our cosmological background. Note that AM and B^ are external nondynamical vectors violating Lorentz symmetry. This result also applies to quantum theory: the excitations 5A and 5B would be seen as the effective particles corresponding to A and B, so that such particles would break Lorentz invariance.

Acknowledgments This work was funded by the Fundagao Portuguesa para o Desenvolvimento.

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SCALAR FIELD MODELS: F R O M T H E P I O N E E R A N O M A L Y TO A S T R O P H Y S I C A L C O N S T R A I N T S

J. PARAMOS* Institute Avenida

Superior Tecnico, Departamento de Fisica, Rovisco Pais 1, 1049-001 Lisboa, Portugal E-mail: [email protected]

In this work we study how scalar fields may affect solar observables, and use the constraint on the Sun's central temperature to extract bounds on the parameters of relevant models. Also, a scalar field driven by a suitable potential is shown to produce an anomalous acceleration similar to the one found in the Pioneer anomaly.

1. Introduction Scalar fields play a fundamental role in contemporary physics, with applications ranging from particle physics to cosmology and condensed matter. In cosmology, these include inflation1, vacuum energy evolving and quintessence models, the Chaplygin gas dark energy-dark matter unification model and dark matter candidates (for a complete list, see Ref. 2). A scalar field has recently been suggested as a possible explanation for the anomalous acceleration measured by the Pioneer spacecraft 3 . On an astrophysical context, a scalar field could be present as, for example, a mediating boson of an hypothetical fifth force. This can be modelled by a Yukawa potential, given by Vy(r) = Ae~mr/r, where A is the coupling strength and m is the mass of the field, setting the range of the interaction Ay = m - 1 . The bounds on parameters A and Ay = m _ 1 can be found in Ref. 4 and references therein. In this work, a scalar field with an appropriate potential is shown to account for the anomalous Pioneer acceleration. Also, the effects of this and other scalar field models on stellar equilibrium is reported.

*Work partially supported by the Fundagao para a Ciencia e Tecnologia, under the grant BD 6207/2001.

298

299 2. Scalar field induced Pioneer anomaly The hypothesis that the Pioneer anomaly is due to new physics has been steadily gaining momentum. Fundamentally, this arises because is seems unfeasible that an engineering explanation, affecting spacecrafts with different trajectories and designs, could lead to similar effects. The lack of a simple explanation could hint that this anomalous acceleration is the manifestation of a new force. Given the broad range of applicability of theories with scalar fields, it is possible that such an entity is responsible for the anomaly 3 . A natural candidate would be the radion, a scalar field present due to oscillations of inter-brane distance in braneworld models. However, this does not account for the phenomena 3 . Fortunately, one can put down a model with a scalar field that doesn't stray away much from the already accepted theories arising from cosmology. Specifically, we assume a scalar field with dynamics ruled by a potential of the form V{) = —A2
(r) = ( - )

vC4F=/rV* .

(2)

The 0 — 0 linearized Einstein's equations reads /"(r) + ; / ' ( r ) = 2 « ( l - ^ ) ^ 2 / ? 2 r - 1

(3)

whose solution is

/M^Vf^-^Ml^) •

W

The resulting acceleration felt by a test body is given by ar =

C

[ZAK

-^V2T

+

[3ACK

V2^- •

(5)

The first term is the Newtonian contribution, and identifying the second with the anomalous acceleration a^ = 8.5 x 10~ 10 ms~2, sets A = 3.26 OA/K = 4.7 x 10 42 m~ 3 . The last term is much smaller than the anomalous acceleration for 4C/r -C 1, that is, for r 3> 6 km; it is also much smaller than the Newtonian acceleration for r
300

to calculate the acceleration of the spacecrafts is negligible, indicating that the anomaly is real, an not due to a misinterpreted light propagation. 3. Variable Mass Particle models The variable mass particle (VAMP) proposal 5 assumes the presence of yet unknown fermions coupled to a dark-energy scalar field with dynamics ruled by a monotonically decreasing potential. Although this has no minima, the coupling of the scalar field (f> to this exotic fermionic matter yields an effective potential of the form Veff(4>) = V(<j>) + Xn^(f>, where n$ is the number density of fermionic VAMPs and A is a Yukawa coupling. In the present work we approach a potential of the quintessence-type form, V{) = UQ(f>~p. As a result, the effective potential acquires a minimum and the related vacuum expectation value (vev) is responsible for the mass term of the exotic VAMP. Since the number density n^ depends on the scale factor a(t), this mass varies on a cosmological timescale. A noticeable shortcoming of VAMP models is the presence of weakly or unconstrained parameters: the relative density fi^o, the scalar field coupling constant and the potential strength. One can attempt to overcome this drawback by assuming that all fermions couple to the quintessence scalar field. This coupling should not substitute the usual Higgs coupling, but merely add a small correction to the Higgs-mechanism induced mass. Since the effects of the quintessence scalar field coupling crucially depend on the particle number density n^,, one expects this variable mass term to play a more relevant role in a stellar environment than in the vacuum. This hints that VAMP model parameters could be constrained from stellar physics observables. 3.1. The polytropic

gas stellar

model

The polytropic gas model assumes an equation of state of the form P — Kpn+lln, where n is the polytropic index, defining intermediate processes between the isothermic and adiabatic thermodynamical cases, and K is the polytropic constant, which depends on the star's mass M and radius R. This model leads to scaling laws for thermodynamical quantities, given by p = Pc0n(O, T = TM) and P = PM)n+1, where pc, Tc and Pc is the density, temperature and pressure at the center of the star 6 . The dimensionless function #(£) depends on the dimensionless variable £, related to the distance to the star's center by r = a£, where a also depends on the star's mass M and radius R. The hydrostatic equilibrium condition enables

301 a differential equation ruling the behavior of the scaling function 0(£), the Lane-Emden equation:

Since the physical characteristics of a star appear only in the definitions of the constants K and a, its stability is independent of these quantities, and different polytropic indexes n label different types of stars. This scaleindependence can be related to the homology symmetry enclosed in the Lane-Emden equation. The boundary conditions of this differential equation are, from the definition p = pc0(£), 9(0) = 1; also, the hydrostatic equilibrium condition implies that \d0/d£\c=0 = 0. The first solar model ever considered corresponds to a polytropic star with n = 3 and was studied by Eddington in 1926. Although somewhat incomplete, this simplified model gives rise to relevant constraints on the physical quantities. The following results are based on the luminosity constraint on the Sun's central temperature, ATC/TC < 0.4%. The central temperature can be computed from Tc = Y„pGM/R, where k is the Boltzmann constant and Yn depends on the star's mass M and radius R, as well as on the dimensionless quantity £i, which is defined through 0(£i) = 0 and signals the surface of the star. 3.2.

Results

Assuming isotropy, a variable mass term leads to both a radial, anomalous acceleration a A = (j>'/<j> < 0 plus a time-dependent drag force an = ~4>I4> < 0 2 . The time-dependent component should vary on cosmological timescales, and can thus can be absorbed in the usual Higgs mass term. Hence, one considers only the perturbation to the Lane-Emden equation given by the radial force. We start by defining the dimensionless scalar field $ = 4>/^, with <j>*c = Pcrit/2nV, where fly is the energy density due to the potential driving <j> and pcrit ~ 9.48 x 10~ 28 g cm~3 is the critical density. Assuming for simplicity a potential with p = 1, we obtain a perturbed Lane-Emden equation

LA.

2M

-0n^-ww)

*"(£) + 7*'(0 £ "'

$(0

(7)

where one has defined the dimensionless quantities U=^i

= 2.12 x H P 6 , C-1 = (n+ i)jV£/< n+1 >W r 1 i /(n+1) ,

(8)

302

withWn = [ 4 7 r ( n + l ) ( f ) 2 ] - 1 . Furthermore, one assumes that the scalar field is only weakly perturbed in relation to the cosmological vev of the effective potential. Denoting this small "astrophysical" contribution as $ a (£), o n e h&s $(£) = fiv/A + $ a (£)Hence, the Klein-Gordon equation becomes 2a 2 Any p{_ /33 c

Kit) + J*'a(Z) >

Pcrit

P

1

HO -

2a2X2n

1-

Pcrit

2A$ a (g) fiy

(9)

inside the star, and

K(0 + fa(S)

2a 2 An v

2a 2 A 2 n v

Pcrit

Pcrit

1-

2A* o (0 fU

(10)

in the outer region. In the above, n$ = ny = 3 m~ 3 is the number density of fermions in the vacuum, /x is the mean molecular weight of Hydrogen. The boundary conditions for the perturbed Lane-Emden are the same as in the unperturbed case. For the scalar field, we assume both $(£) and its derivative vanish beyond the Solar System (about 105 AU). Following Ref. 2, one gets that the maximum deviation ATC/TC = 2.82 x 1 0 - 8 occurs for fiy = 0.7, A = 2.82 x 10~ 14 . Hence, the luminosity constraint is always respected as long as one assumes the bound arising from the assumption that the "astrophysical" component of the scalar field is much smaller than its cosmological vev, A < 1 0 - 1 4 . 4. Yukawa p o t e n t i a l induced p e r t u r b a t i o n One now looks at the hydrostatic equilibrium equation with a Yukawa potential which, after a small algebraic manipulation, implies the perturbed Lane-Emden equation 1

±M = -0"

1 + Ae-iV*1 --yACn—e at, e % dn where we have defined the dimensionless quantities Cn _ =

-7f/Sl

(11)

1/2

fn+l\

j y ; / ( » + i ) ^ i - n ) / J ( n + i ) , >y = mR (12) ; and £i signals the surface of the star (more accurately, a surface of zero temperature, but the difference is negligible. It can be shown that the boundary conditions are unaffected by the perturbation 2 . One can study the variation of the central temperature as a function of A s m~l and A, and constraint these so that ATC/TC < 0.4%. The 47T

303

i
10:1

io 1

IGS'

io 5

107

IG 9

to11

io 13

Figure 1. Exclusion plot for the relative deviation from unperturbed central temperature Tc, for A ranging from I O - 3 to 1 0 - 1 , and 7 from 1 0 " 1 to 10 (tip at the top), superimposed on the available bounds.

parameters were chosen for Yukawa interactions in the range O.li? < Ay < 10.R; the Yukawa coupling A was chosen so that the variation of Tc is of the same order O(10 - 4 )) as the luminosity constraint. Numerical integration of Eq. 11 is then used to derive the exclusion plot of Figure 1, superimposed on the accepted bound 4 . Notice that the central temperature is not precisely known and it is clear that constraining its uncertainty below 10 ~ 4 would yield a larger exclusion region in the parameter space.

4 . 1 . The Pioneer

anomaly

Following a similar procedure to the one depicted in the above two sections, one can prove that a constant, anomalous acceleration a A inside the Sun yields a relative deviation of the central temperature which scales linearly with a A as STC ~ a^/a©, where a© = 274 m.s~22. Thus, the bound STC < 4 x 10~ 3 is satisfied for values of this constant anomalous acceleration up to a,Max ~ lO _ 4 a0. The reported value is then well within the allowed region, and has a negligible impact on the astrophysics of the Sun.

304

5. Conclusion In this work we have developed a study of the impact of some exotic physics models on stellar equilibrium, enabling the extraction of constraints on the relevant parameters of the theories. Also, a model exhibiting a scalar field with a attractive quintessence-like potential is discussed, which can account for the anomalous acceleration felt by the Pioneer 10/11, Ulysses and Galileo spacecrafts. References 1. 2. 3. 4. 5. 6. 7. 8. 9.

See e.g. A. Linde, hep-th/0402051 and K.A. Olive, Phys. Rev. 190 (1990) 307. O. Bertolami, J. Paramos Phys. Rev. 71, (2005) 23521. O. Bertolami, J. Paramos, Class. Quantum Gravity 21 (2004) 3309. "The search for non-Newtonian gravity", E. Fischbach, C.L. Talmadge (Springer, New York 1999). G.W. Anderson, S.M. Carroll, astro-ph/9711288. "Textbook of Astronomy and Astrophysics with Elements of Cosmology", V.B. Bhatia (Narosa Publishing House, Delhi 2001). "Theoretical Astrophysics: Stars and Stellar Systems", T. Padmanabhan (Cambridge University Press, Cambridge 2001). J.N. Bahcall, Phys. Rev. D33 (2000) 47. J.D. Anderson, P.A. Laing, E.L. Lau, A.S. Liu, M.M. Nieto and S.G. Turyshev, Phys. Rev. D65 (2002) 082004, Phys. Rev. Lett. 81 (1998) 2858.

B R A N E W O R L D S , CONFORMAL FIELDS A N D D A R K ENERGY

RUI NEVES Departamento

de Fisica, Faculdade de Ciencias e Tecnologia Universidade do Algarve & Centra Multidisciplinar de Astrofisica-CENTRA Campus de Gambelas, 8005-139 Faro, Portugal E-mail: [email protected]

In the Randall-Sundrum scenario we analize the dynamics of a spherically symmetric 3-brane when matter fields propagate in the bulk. For a well defined class of conformal fields of weight -4 we determine a new set of exact 5-dimensional solutions which localize gravity in the vicinity of the brane and are stable under radion field perturbations. Geometries which describe the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic dark energy on the brane are shown to belong to this set.

1. Introduction In the Randall-Sundrum (RS) scenario 1,a the visible Universe is a 3-brane world of a Z% symmetric 5-dimensional (5D) anti-de Sitter (AdS) space. In the RSI model * there is a compact fifth dimension and two brane boundaries. The gravitational field is localized near the hidden positive tension brane and decays towards the visible negative tension brane. In this model the hierarchy problem is reformulated as an exponential hierarchy between the weak and Planck scales : . In the RS2 model 2 there is a single positive tension brane in an infinite fifth dimension. Then gravity is bound to the positive tension brane now interpreted as the visible brane. At low energies the theory of gravity on the observable brane is 4dimensional (4D) general relativity and the cosmology may be FriedmannRobertson-Walker 1 - 1 0 . In the RSI model this is only possible if the radion mode is stabilized using for example a bulk scalar field 3-6'9>10. Gravitational collapse was also analyzed in the RS scenario 11 - 16 . However, an exact 5D solution representing a stable black hole localized on a 3-brane has not yet been discovered. So far, the only known static black holes localized

305

306

on a brane remain to be those found for a 2-brane in a 4D AdS space 12 . A solution to this problem requires the non-singular localization of both gravity and matter n > 1 3 - 1 6 and could be connected to quantum black holes on the brane 15 . This is an extra motivation to look for 5D collapse solutions localized on a brane. In addition, the effective covariant GaussCodazzi approach 17 ' 18 has permitted the discovery of many braneworld solutions which have not yet been associated with exact 5D spacetimes 19_22

In this paper we continue the research on the dynamics of a spherically symmetric RS 3-brane when 5D conformal matter fields propagate in the bulk 16,23 (see also 2 4 ) . In our previous work 16,23 we have found a new class of exact 5D dynamical solutions for which gravity is localized near the brane by the exponential RS warp. These solutions were shown to describe on the brane the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter. However, the density and pressures of the conformal bulk fluid increase with the coordinate of the fifth dimension. In the RS2 model this generates a divergence at the AdS horizon as in the Schwarzschild black string solution n . In the RSI model this is not a problem. However, the solutions turn out to be unstable under radion field perturbations 25 . In this work we report on a new set of exact 5D braneworld solutions which have a stable radion mode and still describe on the observable brane the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter.

2. 5D Einstein Equations and Conformal Fields On the 5D RS orbifold the non-factorizable metric consistent with the Z2 symmetry in z and with 4D spherical symmetry on the brane corresponds to the 5D line element d~s\ = fl2 (dz2 - e2Adt2 + e2Bdr2 + R2dSl\), where Q = Cl(t, r, z), A — A(t, r, z), B = B(t, r, z) and R = R(t, r, z) are Z2 symmetric functions. R(t,r,z) represents the physical radius of the 2-spheres and fl is the warp factor which defines a global conformal transformation on the metric. The classical dynamics is defined by the 5D Einstein equations, Gl = -K2 J A B ^ + ~ ~ [A<5 (z - z0) + X'6 (z - z'„)] ( ^ - 8%5*) - ?„"} ,(1) where Ag is the negative bulk cosmological constant, A and A' are the brane tensions, K\ — 87r/Mf with M 5 the fundamental 5D Planck mass and T£

307

is the stress-energy tensor associated with the matter fields. In 5D T£ is conserved, V M 7 ^ = 0. Let us consider the special class of conformal bulk matter defined by T£ = Q,~2T^ and assume that T£ depends only on t and r. The conformal stress-energy tensor T^u may be separated in two sectors T^" and £/"'"' with the same weight s, t ^ = f^ + U^ where f ^ = fi'T"" and U>iU = Q^SJJUV Assuming that T^ and U£ are conserved tensor fields, A = A(t, r), B = B(t, r), R = R(t, r) and fi = Cl(z) we obtain

Gba = 4Tba,

G\ = 4Tl

VaT6a = 0, VQf/6a = 0,

Gtor2(dznf + 4fl2AB {zn-^n

+ 4n

2

(2)

= k2Ul

1

(3) b

{ A B + n - \\6(z - z0) + x'6(z - z'0)}}}s

2 b

a

= k u a, (4)

where the latin indices represents the 4D coordinates t, r, 9 and
-2pT + 2p5 = 0,

(5)

where p, pr, px and p$ must be independent of z but may be functions of t and r. The bulk matter is, however, inhomogeneously distributed along the fifth dimension because the physical energy density, p(t, r, z), and pressures, p(t,r,z), are related to p{t,r) and p(t,r) by the scale factor H,~2(z). Note also that the warp depends on the conformal bulk fields only through Up. So the role of U" is to influence how the gravitational field is warped around the branes. On the other hand T^ determines the dynamics on the branes. In the RSI model the two branes have identical cosmological evolutions and gravity will be localized on the Planck brane and not on the visible one. 3. Exact 5D Warped Solutions The dynamics of the AdSs braneworlds is defined by the solutions of equations (2) to (5). Let us first solve the warp equations (3) and (4). If ps = 0 then U£ = 0 and we obtain the usual RS warp equations. A solution is the exponential RS warp 1,a . Using the coordinate y related to z by z = leyll for y > 0 we find

nG/) = fWi/) = e-i*i/',

(6)

308

where / is the AdS radius given by I = I / ^ - A S K 2 / ^ . If ps is non-zero then there is a new set of warp solutions to be considered. Integrating Eq. (3) and taking into account the Z2 symmetry we find

n(„) = e -i»i/»A

+

^e2i»i/A

(7)

This set of solutions which depends on the 5D pressure p$ must also satisfy Eq. (4) which contains the Israel conditions. This may only happen if the brane tensions A and A' are given by x _

b l 4AB / K 2 1 4 . _£§_ '

,/ A

O L 4ABe ; „ 2 1 , _ g s _ 27rr c // '

,«N W

where r c is the RS compactification scale. The conformal factorfi(y)defines how the gravitational field is warped around the brane. To find the dynamics on the brane we need to consider solutions of Eq. (2) when the diagonal bulk matter T£ satisfies Eq. (5). For inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter such solutions were determined in Refs. 16 and 23 . The latter describes the dynamics on the brane of dark energy in the form of a polytropic fluid. The diagonal conformal matter may be defined by P = Pp,Pr + I P P " = 0 , P T = Pr,P5 = - (p P + 3?7/OpQ) / 2 , where p P defines the polytropic energy density and the parameters (a, 77) characterize different polytropic phases. For — 1 < a < 0 the fluid is in its generalized Chaplygin phase (see also 2 6 ) . The 5D polytropic solutions are 2 3 ds\ = fi2

-dt2+S2[^^+r2dnl

+ dy2,

(9)

where the Robertson-Walker brane scale factor S satisfies ,2

S' =

-*+^(i+s^r"-

(io)

4. Radion Stability To analyze how these solutions behave under radion field perturbations we apply a saddle point expansion procedure based on the action 27,28 j ^ u s write the most general metric consistent with the Z2 symmetry in y and with 4D spherical symmetry on the brane in the form ds2 = o?ds\ + b2dy2 with ds2 = -dt2 + e2Bdr2 + R2dD.2- The metric functions a = a(t,r,y), B = B(t, r,y), R = R(t, r, y) and b = b(t, r, y) are Z2 symmetric. Now a is

309

the warp factor and 6 is related to the radion field. The 5D dynamical RS action is given by

zdyVHJ \7^-AB--7^=1X6 2K

/ "

5

\/555

""

(y) + X'S (y - nrc)} + LB } .(11)

Our braneworld backgrounds correspond to the metric functions 6 = 1 , B = B(t,r), R = R(t, r) and a = Q(y). To calculate the radion potential we consider the dimensional reduction of (11). Using the metric with a(t,r,y) = Vt{y)e-^^ and b(t,r) = e 0 ^ we obtain in the Einstein frame 25 5 = | JxyTgi

( ^

- \Vc7Vd734cd

- V^j ,

(12)

where 7 = [3/(K4y/2/3) is the canonically normalized radion field. The function V — V{-)) is the radion potential and it is given by 2 K\5

,

3 f dyQ.2{dvnf

+ 2 /dytfdlQ

+xf

+X2 J dyn4 [XS (y) + X'S (y - 7rrc)),

dyQ4 (AB - L B ) (13)

where the field x i s defined The integration in the fifth dimension is performed in the interval [—7rrc,7rrc] and that we have chosen J dyQ2 = K5/K4. To analyze the stability of the AdSs braneworld solutions we consider a saddle point expansion of the radion field potential V. If ps = 0 then Q = flRS. The critical extremum corresponding to our braneworlds is x = 1 25 . Stable solutions must be associated with a positive second variation of the radion potential. If the equation of state of the conformal bulk fields is independent of the radion perturbation then for x = 1 the second variation is negative and so the corresponding braneworlds are unstable 25 . If the equation of state is kept invariant under the radion perturbations it is possible to find stable solutions at x — 1 if the warp is changed. Indeed, the new relevant warp functions are given in Eq. (7) and stability exists for a range of the parameters if p5 < 0. As an example consider the interval 4ABe _27rro//i < ps < 0 which corresponds to a brane configuration with A > 0 and A' < 0. Then the stability conditions are I > 3r c , 4A B e _27rr < :/ ' < p5
310 5. Conclusions In this paper we have analized exact 5D dynamical solutions with gravity localized near the brane which are associated with conformal bulk fields of weight -4 and describe the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter on the brane. We have discussed their behaviour under radion field perturbations and shown that they are extrema of the radion potential. We have also shown that if the equation of state characterizing the conformal fluid is independent of the perturbation then the radion may be stabilized by a sector of the conformal fields while another sector generates the dynamics on the brane. Stabilization requires a bulk fluid with a constant negative pressure and involves new warp functions. On the brane these solutions also describe the dynamics of inhomogeneous dust, generalized dark radiation and homogeneous polytropic matter. Whether gravity is suficiently localized on the brane is an open problem for future research. Acknowledgments We would like to thank the financial support of Fundagao para a Ciencia e a Tecnologia (FCT) and Fundo Social Europeu (FSE) under the contract SFRH/BPD/7182/2001 (III Quadro Comunitdrio de Apoio) as well as of Centro Multidisciplinar de Astrofisica (CENTRA). References 1. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999). 2. L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999). 3. W. D. Goldberger and M. B. Wise, Phys. Rev. D. 60, 107505 (1999); Phys. Rev. Lett. 83, 4922 (1999); Phys. Lett. B 475, 275 (2000). 4. N. Kaloper, Phys. Rev. D 60, 123506 (1999); T. Nihei, Phys. Lett. B 465, 81 (1999); C. Csaki, M. Graesser, C. Kolda and J. Terning, Phys. Lett. B 462, 34 (1999); J. M. Cline, C. Grojean and G. Servant, Phys. Rev. Lett. 83, 4245 (1999). 5. P. Kanti, I. I. Kogan, K. A. Olive and M. Pospelov, Phys. Lett. B 468, 31 (1999); Phys. Rev. D 61, 106004 (2000). 6. O. DeWolf, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D 62, 046008 (2000). 7. J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000). 8. S. Giddings, E. Katz and L. Randall, J. High Energy Phys. 03, 023 (2000). 9. C. Csaki, M. Graesser, L. Randall and J. Terning, Phys. Rev. D 62, 045015 (2000). 10. T. Tanaka and X. Monies, Nucl. Phys. B582, 259 (2000).

311 11. A. Chamblin, S. W. Hawking and H. S. Reall, Phys. Rev. D 61, 065007 (2000). 12. R. Emparan, G. T. Horowitz and R. C. Myers, J. High Energy Phys. 0001, 007 (2000); J. High Energy Phys. 0001, 021 (2000). 13. P. Kanti, K. A. Olive and M. Pospelov, Phys. Lett. B 481, 386 (2000); T. Shiromizu and M. Shibata, Phys. Rev. D 62, 127502 (2000). 14. P. Kanti and K. Tamvakis, Phys. Rev. D 65, 084010 (2002); P. Kanti, I. Olasagasti and K. Tamvakis, Phys. Rev. D 68, 124001 (2003). 15. R. Emparan, A. Fabbri and N. Kaloper, J. High Energy Phys. 0208, 043 (2002). 16. R. Neves and C. Vaz, Phys. Rev. D 68, 024007 (2003). 17. B. Carter, Phys. Rev. D 48, 4835 (1993); R. Capovilla and J. Guven, Phys. Rev. D 51, 6736 (1995); Phys. Rev. D 52, 1072 (1995). 18. T. Shiromizu, K. I. Maeda and M. Sasaki, Phys. Rev. D 62, 024012 (2000); M. Sasaki, T. Shiromizu and K. I. Maeda, Phys. Rev. D 62, 024008 (2000). 19. J. Garriga and M. Sasaki, Phys. Rev. D 62, 043523 (2000); R. Maartens, D. Wands, B. A. Bassett and I. P. C. Heard, Phys. Rev. D 62, 041301 (2000); H. Kodama, A. Ishibashi and O. Seto, Phys. Rev. D 62, 064022 (2000); D. Langlois, Phys. Rev. D 62, 126012 (2000); C. van de Bruck, M. Dorca, R. H. Brandenberger and A. Lukas, Phys. Rev. D 62, 123515 (2000); K. Koyama and J. Soda, Phys. Rev. D 62, 123502 (2000). 20. N. Dadhich, R. Maartens, P. Papadopoulos and V. Rezania, Phys. Lett. B 487, 1 (2000); N. Dadhich and S. G. Ghosh, Phys. Lett. B 518, 1 (2001); C. Germani and R. Maartens, Phys. Rev. D 64, 124010 (2001); M. Bruni, C. Germani and R. Maartens, Phys. Rev. Lett. 87, 231302 (2001). 21. R. Maartens, Phys. Rev. D 62, 084023 (2000). 22. R. Neves and C. Vaz, Phys. Rev. D 66, 124002 (2002). 23. R. Neves and C. Vaz, Phys. Lett. B 568, 153 (2003). 24. E. Elizalde, S. D. Odintsov, S. Nojiri and S. Ogushi, Phys. Rev. D 67, 063515 (2003); S. Nojiri and S. D. Odintsov, JCAP 06, 004 (2003). 25. R. Neves, in TSPU Vestnik, edited by S. D. Odintsov, vol. 7 (2004) p. 94. 26. M. C. Bento, O. Bertolami and S. S. Sen, Phys. Rev. D 66, 043507 (2002); Phys. Rev. D 67, 063003 (2003); Phys. Rev. D 70, 083519 (2004); M. C. Bento, O. Bertolami, N. M. C. Santos and S. S. Sen, Phys. Rev. D 71, 063501, 2005. 27. R. Hofmann, P. Kanti and M. Pospelov, Phys. Rev. D 63, 124020 (2001); P. Kanti, K. A. Olive and M. Pospelov, Phys. Lett. B 538, 146 (2002). 28. S. M. Carroll, J. Geddes, M. B. Hoffman and R. M. Wald, Phys. Rev. D 66, 024036 (2002).

S U N A N D STARS AS COSMOLOGICAL TOOLS: P R O B I N G SUPERSYMMETRIC DARK MATTER

ILIDIO LOPES* Centro

de Geofisica de Evora, Departamento de Fisica, Universidade de Evora, Colegio Luis Antonio Verney, 7002-554 Evora - Portugal CENTRA, Departamento de Fisica, Instituto Superior Tecnico Av. Rovisco Pais 1, 1049-001 Lisboa - Portugal Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, 0X1 3RH Oxford - United Kingdom

Observational cosmology has shown that a quarter of the matter of the universe is composed by nonbaryonic dark matter. This dark matter may well constitute halos of massive and annihilating particles, weakly interacting and if so, these particles would be trapped in the interior of stars like our Sun. Whence solar evolution, in combination with helioseismic data, can provide clues as to the existence of nonbaryonic dark matter and some of their intrinsic properties. The current status of the solar standard model opens the way for using the current solar observations, namely, the neutrino flux detections and the solar acoustic data to test and probe the new physics behind the standard model of particle physics and cosmology. In this article we make a brief presentation of the physics behind the presence of dark matter inside stars.

1. The present Sun: The solar standard model The Sun is a unique star for research because its proximity allows a superb quality of solar data, enabling precision measures of its luminosity, mass, radius and chemical composition. The Sun therefore naturally becomes a privileged target for testing stellar evolution theory. In recent years, different groups around the world have produced solar models in the framework of classical stellar evolution, taking into account the best known physics as well as all the available observational seismic data. This has led to the determination of a well-established model for the Sun, the so-called standard solar model [17, 4, 12, 14, 3], for which the acoustic modes are in *Work partially supported by grant pocti/fis/57568/2004 of the Fundagao para a Ciencia e Tecnologia.

312

313 WIMPs

i a o.ooo

jfe-u^lfitel

^^t^Pf^r

- 1 0 - 2 7 em 9 /sec

+ II. ,

^

"3Hfcr

y . -0.001

+

solar radius

Figure 1. The relative differences between the square of the sound speed (a) and the density (b) of the solar model embedded in dark matter halos and the standard solar model. The continuous curves correspond to models of WIMPs with masses mx of 15 GeV, 30 GeV and 60 GeV with annihilation cross-section of 10 7cm /s, and scalar scattering cross-section
314 some of the basic properties of the microscopic physics of the solar interior [e.g. 15]. This solar model has established considerable consensus among the different research groups, concerning the predictions of the solar neutrino fluxes, and has unambiguously helped define the difference between the theoretical predictions and the experimental results. In the previous three decades, solar neutrino experiments have measured fewer neutrinos than were predicted by the solar models. One explanation for the deficit is the transformation of the Sun's electron-type neutrinos into other active flavors. Recently, the Sudbury Neutrino Observatory (SNO) has measured the 8B solar neutrinos. The results obtained establish direct evidence for the non-electron flavor component in the solar neutrino flux and yield the first unequivocal determination of the total flux of SB neutrinos produced by the Sun [2, 3]. The combined effort between helioseismology and stellar astrophysics has contributed to strongly constrain the internal structure of the present Sun. Indeed, among the different seismic diagnostics that the solar model is tested against, the radial profile of the square of the sound speed is known with a precision as high as 0.3%, and in particular in the nuclear region this precision goes up to 0.2% (see Fig 1). This progress has been achieved through a detailed description of the microscopic physics, such as the nuclear reaction rates, the equation of state, the coefficient of opacities and the gravitational settling of chemical elements. More recently, the macroscopic mixing induced by differential rotation, seems to present a determining role in the evolution of the star. These physical processes take place in the thin shear layer, located between the radiative interior and the convection zone, the so-called tachocline [5]. The standard stellar evolution of the Sun assumes that the star is in hydrostatic equilibrium, is spherically symmetric and that the effects of rotation and magnetic fields are negligible. The present solar structure and evolution are computed starting from an initially homogeneous star of a given composition. The main physical inputs are as follows. The microphysics data are obtained from different sources: the solar composition [7], the OPAL96 tables [8], the equation of state [13], the nuclear reaction rates [1], the prescription for the treatment of the nuclear screening rates [10], and the microscopic diffusion coefficients suggested by [9]. The present structure of the Sun is obtained by evolving a initial star from the pre-main sequence, around 0.05 Gyr from the ZAMS, until its present age, 4.6 Gyr. The present solar model is obtained by choosing the initial Helium content, as well as the mixing length parameter of convection that best predicts

315 the present solar luminosity and radius. The helioseismology requires that the present structure of the Sun, including the global quantities, should be determined with a very high precision. In particular, the calibration of the solar radius is done with a precision of 10~ 5 . If our understanding of the basic equilibrium structure of the present Sun seems quite-well established, the same cannot be said about the dynamical processes acting in its interior, such as the turbulent convection, the transport of angular momentum, and the mechanism excitation of acousticgravity waves. These physical processes are particularly critical in regions of the star that somewhat regulate their evolution but also in regions where important dynamical processes are taking place in a much more short scale, from some minutes to several years, which is the case of the base of the convection zone, where the magnetic field is believed to be generated [6], and in the super-adiabatic region where the modes are believed to be excited [11]. If the stellar evolution theory still presents some unsolved problems, which is particularly true in the case of the Sun, it is fair to say that we have obtained a good understanding of the basic processes behind the evolution of the Sun and stars. This will allow us to use the Sun and stars as an instrument to test new physical ideas, namely to research the particle candidates for dark matter.

2. Helioseismology and Dark Matter The knowledge of the Sun has improved with the advent of helioseismology. Since different modes of oscillations penetrate different depths in its interior, the measured frequencies impose severe constraints on the models of the present Sun, ruling out most of those which have been put forward to reconcile the difference between the theoretical and the observed spectra [17]. A similar strategy is used in the research for dark matter. To obtain an understanding of the physical processes engaged in stellar interiors by SUSY particles, we need different solar models for a wide range of particles with different properties, and to consider different abundances in its interior. The measured oscillation frequencies by the three helioseismology instruments on board the Solar and Heliospheric Observatory (SOHO) have put severe constraints on the variation of sound speed with radius and also with density, especially in the central regions which have the greatest importance for determining the impact of dark matter in stellar evolution: they also enhance our knowledge of the nature of the dark

316

-44

-I?

-40 -H8 K>9,(, ".„„• IO"l'l

-36

-34

m x (GeV)

Figure 2. Relative variation of the luminosity in the core of the Sun, due to the concentration of WIMPs in the solar core, as a function of scalar scattering cross section and annihilation cross section (a), and as a function of WIMP mass and scattering cross-section (b). The iso-curves represent the decimal logarithm of the ratio of the luminosity of WIMPs against the Sun's luminosity in the inner core, within 5% of the solar radius. The computation was made using the structure of the present Sun. The solid curves represent the current and future experimental bounds placed by different experiments. m a t t e r particles. It follows from Fig. 1 t h a t the sound speed and density in the interior of the Sun is presently known with a precision of much less t h a n a few per cent. This level of accuracy in constraining the solar in-

317 terior has been achieved by a systematic study of the differences between the acoustic spectrum obtained from helioseismology experiments and the theoretical spectrum. Presently, this difference is less than several micro Hertz for almost all of the three thousand acoustic modes that probe the interior of the Sun. Nevertheless, a higher accuracy in probing the structure of the nuclear region is expected in the coming years, provided by more accurate measurements of acoustic modes, but also, for the first time, by the detection and measurement of gravity modes [16]. Even if the answers to some questions about the stellar physics are still unclear, it seems very likely that there is a class of annihilating weak interacting massive particles ( 1 0 - 2 7 cm3/sec) and relatively small masses (60 GeV) that is already excluded by the present seismic results. This type of analysis will enable comparisons to be made with a range of theoretical models built on the basis of different assumptions in the nature of interacting massive particles, leading to improvements in the origin and nature of dark matter. Even in the cases of distant stars, from which the data will be of relatively low quality compared with that from the Sun, measurements of oscillation frequencies of a number of stars in our Galaxy can be used to calibrate, constrain, test and improve our knowledge about the distribution of dark matter in the Milky Way. The thermodynamic structure of the interior of the Sun, namely in the nuclear region, is presently known with a precision of much less than a few per cent. In particular, the luminosity in the solar core is known with a precision of one part in thousand. In the coming years, it is very likely that the new seismic data available from the SOHO experiments, will allow us to obtain a solar model with seismic constraints by as much as one part in a hundred thousand. Alternatively, as it follows from Fig. 2, the contribution of the WIMPs luminosity for the total luminosity of the Sun's core is above the accuracy of seismic data. In such conditions, the Sun can and should be used as a probe for dark matter in the solar neighborhood, spanning the parameter space of SUSY particle theory (cf. Fig. 2). This order of magnitude in the luminosity produced by the solar core can be tested through seismological data, leading to the prediction of the annihilating neutrino flux with an accuracy not available with the present SUSY particle models.

318 References 1. E. G. et al. Adelberger. Solar fusion cross sections. Reviews of Modern Physics, 70:1265-1291, October 1998. 2. Q. R. et al. Ahmad. Measurement of the Rate of j / e + d : p + p + e~ Interactions Produced by 8 B Solar Neutrinos at the Sudbury Neutrino Observatory. Physical Review Letters, 87:071301—h, August 2001. 3. J. N. Bahcall. High-energy physics: Neutrinos reveal split personalities. Nature, 412:29-31, July 2001. 4. J. Christensen-Dalsgaard, W. Dappen, S. V. Ajukov, E. R. Anderson, H. M. Antia, S. Basu, V. A. Baturin, G. Berthomieu, B. Chaboyer, S. M. Chitre, A. N. Cox, P. Demarque, J. Donatowicz, W. A. Dziembowski, M. Gabriel, D. 0 . Gough, D. B. Guenther, J. A. Guzik, J. W. Harvey, F. Hill, G. Houdek, C. A. Iglesias, A. G. Kosovichev, J. W. Leibacher, P. Morel, C. R. Proffitt, J. Provost, J. Reiter, E. J. Rhodes, F. J. Rogers, I. W. Roxburgh, M. J. Thompson, and R. K. Ulrich. The Current State of Solar Modeling. Science, 272:1286-+, 1996. 5. J. R. Elliott and D. O. Gough. Calibration of the Thickness of the Solar Tachocline. ApJ, 516:475-481, May 1999. 6. D. Gough. Some theoretical remarks on solar oscillations. In Nonradial and Nonlinear Stellar Pulsation, pages 273-299, 1980. 7. N. Grevesse, A. Noels, and A. J. Sauval. Standard Abundances. In ASP Conf. Ser. 99: Cosmic Abundances, pages 117-+, 1996. 8. C. A. Iglesias and F. J. Rogers. Updated Opal Opacities. ApJ, 464:943+, June 1996. 9. G. Michaud and C. R. Proffitt. Particle transport processes. In ASP Conf. Ser. 40: IAU Colloq. 137: Inside the Stars, pages 246-259,1993. 10. H. E. Mitler. Thermonuclear ion-electron screening at all densities. I Static solution. ApJ, 212:513-532, March 1977. 11. R. Nigam and A. G. Kosovichev. Source of Solar Acoustic Modes. ApJ, 514:L53-L56, March 1999. 12. J. Provost, G. Berthomieu, and P. Morel. Low-frequency p- and g-mode solar oscillations. 353:775-785, January 2000. 13. F. J. Rogers, F. J. Swenson, and C. A. Iglesias. OPAL Equation-ofState Tables for Astrophysical Applications. ApJ, 456:902—h, January 1996. 14. S. Turck-Chieze, S. Couvidat, A. G. Kosovichev, A. H. Gabriel, G. Berthomieu, A. S. Brun, J. Christensen-Dalsgaard, R. A. Garcia, D. O. Gough, J. Provost, T. Roca-Cortes, I. W. Roxburgh, and R. K.

319 Ulrich. Solar Neutrino Emission Deduced from a Seismic Model. ApJ, 555:L69-L73, July 2001. 15. S. Turck-Chieze, W. Dappen, E. Fossat, J. Provost, E. Schatzman, and D. Vignaud. The solar interior. Phys. Rep., 230:57-235, 1993. 16. S. Turck-Chieze, R. A. Garcia, S. Couvidat, R. K. Ulrich, L. Bertello, F. Varadi, A. G. Kosovichev, A. H. Gabriel, G. Berthomieu, A. S. Brun, I. Lopes, P. Palle, J. Provost, J. M. Robillot, and T. Roca Cortes. Looking for Gravity-Mode Multiplets with the GOLF Experiment aboard SOHO. ApJ, 604:455-468, March 2004. 17. S. Turck-Chieze and I. Lopes. Toward a unified classical model of the sun - On the sensitivity of neutrinos and helioseismology to the microscopic physics. ApJ, 408:347-367, May 1993.

ZEPLIN III: A XENON DETECTOR FOR WIMP SEARCHES HENRIQUE ARAUJO, on behalf of the ZEPLIN Collaboration1 Blacken Laboratory, Imperial College London, SW7 2BW, UK. ZEPLIN III is a two-phase xenon detector soon to be deployed underground at the Boulby mine to search for dark matter in the form of Weakly Interacting Massive Particles (WIMPs). It exploits both scintillation and ionisation signals produced in liquid xenon in order to discriminate between gamma-like interactions and the nuclear recoils expected from the scattering of WIMPs off Xe atoms. This discrimination of particle species, together with the low radioactivity of its components, should extend the search for WIMP-nucleon interactions down to a spin-independent cross-section of 10 s pb.

1. Introduction ZEPLIN III is a two-phase (liquid/gas) xenon experiment which aims to detect galactic dark matter in the form of WIMPs [1]. It operates on the principle that electron and nuclear recoils, produced in liquid xenon by different particle species, generate different amounts of scintillation light and ionisation charge for a given deposited energy. WIMPs are expected to scatter elastically off Xe atoms in the same way as neutrons. The resulting nuclear recoils produce a different signature to electron recoils caused by gamma-ray interactions, which are the dominant source of background in the target. Two-phase emission detectors using condensed noble gases were proposed some three decades ago [2]. ZEPLIN II, a two-phase system operating at low fields [3], is presently being installed underground at the Boulby mine (North Yorkshire, UK). ZEPLIN III follows extensive R&D into two-phase operation at high electric fields [4,5]. In these so-called hybrid detectors, the vacuum ultraviolet (VUV) photons produced by the scintillation and, indirectly, the ionisation channels are measured by the same array of photomultiplier tubes (PMTs) immersed in the liquid. The ability to identify rare nuclear recoils also relies on a low-background construction using radio-pure materials, a thin-slab geometry which improves light collection and ensures uniform electric fields,

Edinburgh University, Imperial College London, ITEP-Moscow, LIP-Coimbra, Rochester University, Rutherford Appleton Laboratory, Sheffield University, Texas A&M, UCLA.

320

321 and capability for 3-dimensional event reconstruction. 2. The ZEPLIN III detector The Xe target and PMT array are located in a cold vessel (-100°C) which sits above a liquid N2 reservoir. Both are mounted inside a vacuum vessel ^0.7 m in diameter and 1.1 m in height, as shown in Fig. 1 (left). High-purity copper is the main construction material, which reduces potential radioactive backgrounds.

Fig.l: Left: Cut-out view of ZEPLIN ID copperwork. Right: Fully-assembled PMT array.

Inside the target chamber, a 40-mm thick disk of liquid xenon is viewed from below by 31 PMTs immersed in the liquid [6]. The assembled array is shown in Fig. 1 (right). The PMTs are biased by a connector-network of copper plates located in the liquid, thus reducing the number of required feedthroughs considerably. A strong electric field (-8 kV/cm) is applied to the target between a top electrode mirror in the gas phase and a wire grid above the PMTs. This field extracts the ionisation from the interaction site into the 5-mm thick gas layer which exists above the liquid. A reverse-field region is created by a second grid located just above the phototubes in order to suppress ionisation signals from the bottom 5 mm of liquid, and so reduce some of the gamma-ray background from the PMTs. Submerging the photomultipliers in the liquid ensures good light collection for the primary scintillation, which is critical given the small number of photoelectrons (phe) expected from WIMP interactions. The simulated light collection efficiency, shown in Fig. 2 (left), is ^3.5 phe/keV in the active region

322 above the cathode grid, except near the field-shaping rings, and is quite constant over the z (vertical) coordinate [7]. This yield would be observed for gammarays in the absence of field; high-field operation reduces the number of VUV photons to -1/3 of the zero-field value -due to partial suppression of. the recombination of the ionisation charge generated at the interaction site [8,9,12]. Lower suppression is expected for nuclear recoils, although these generate only a fraction (-20%) of the prompt scintillation light relative to an electron of the same energy [10-12]. The high field in the liquid allows ionisation to be extracted from the primary tracks. More charge can be collected from electron interactions than from nuclear recoil ones, owing to the higher ionisation density around the primary track in the latter case [13]. Any charge which escapes recombination will drift upward in the liquid and, at such high fields, can be extracted into the gas phase with nearly unity efficiency [14]. Here, the emitted electrons acquire enough energy to produce excitation and electroluminescence of gaseous xenon, resulting in ample generation of 175 nm photons. At nominal field (17 kV/cm in the gas) a single electron extracted from the interaction site can generate 500 VUV photons in the 5 mm gas layer in this proportional scintillation regime. This gives an average ~26 phe detected in the PMTs [8,15] — highlighting the sensitivity which can be achieved for the ionisation channel. Figure 2 (right) shows the light collection efficiency expected across the gas phase. anoctemiiror fls > ftquxi iwja»* j j

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20 40 60 80 100 120 140 160 180

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The two signals generated by every interaction in the active region provide both discrimination of the type of interacting particle and 3-dimensional information of the interaction point. When combined, these features allow excellent • discrimination of gamma-ray and other reducible backgrounds. The ratio between the electroluminescence (52) and the prompt scintillation (SI) signals will be much higher for gamma-ray interactions than for nuclear recoils. An analysis of the discrimination ability of ZEPLIN III [16] has shown that a gamma-ray rejection fraction of 10"5 can be achieved whilst maintaining good detection efficiency for nuclear recoils down to a few SI photoelectrons, as shown in Fig. 3. This suggests- that a (zero-field) threshold of ~*2 keV in visible energy (lOkeV recoils) is feasible. Also, we note that ZEPLIN III is sensitive down to 1 electron extracted from the liquid. Since the system is triggered by the larger S2 signals, the effective threshold for SI is therefore quite low.

324

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Primary photoelectrons Fig. 3: Simulation of particle discrimination in ZEPLIN III [16]: scatter plot of the primary and secondary harvest of photoelectrons from electrons (upper distribution) and nuclear recoils (lower). The latter interactions are assumed to be a-like with a quenching factor of 0.2. The horizontal segments follow a 10"5 rejection boundary.

Three-dimensional information allows the identification of peripheral events and the fiducialisation of the active volume. Particles interacting in outlying regions, where the electric field may not be uniform, can therefore be identified. Small energy deposits from a, p and nuclear recoils arising from radioactive contamination of detector surfaces can also be excluded. The (x,y) coordinates of the interaction can be reconstructed from S2 alone. Sub-centimetre accuracy (FWHM) can be obtained for small signals (a few ionisation electrons extracted from the liquid). The z coordinate is calculated from the time difference between SI and 52; the saturation drift velocity at high field is approximately 3 mm/us in liquid xenon [17]. A Xe fiducial mass in excess of 6 kg should be achieved, corresponding to a disk 3.5 cm thick and ==28 cm in diameter. 3. WIMP-nucleon sensitivity PMT radioactivity is the dominant internal source of background in the detector. Gamma-rays from the 238U and 232Th decay chains and from 40K were measured at just over 105 /day/PMT, setting a detector trigger rate of a few events per second. An interaction rate of around lOdru (1 dru=l evt/kg/day/keV) is expected in the low-energy region in the active volume. A 10"5 rejection efficiency should remove this major background. Neutrons from (a,n) reactions

325

in the PMT glass will cause an irreducible background of nuclear recoils. This is estimated at ~10"3 dru at 2keV threshold (lOkeV recoil energy), totalling =15 single recoil events per year above threshold in a 6 kg fiducial mass. A judicious choice of the materials and techniques used in the detector construction ensures that other potential neutron and electron/gamma backgrounds are kept to a minimum. Xenon contamination with 85Kr, a p emitter with T]/2=10.16 yr, should be minimised with the sourcing of xenon some four decades old (present equivalent Kr concentration ~10ppb Kr). External backgrounds, mainly gamma-rays and neutrons emanating from the Boulby rock, are mitigated by extensive passive shielding. For this purpose, the detector is enclosed in a 15 cm-thick lead castle, containing a further 30 g/cm2 of hydrocarbon material for neutron moderation. The neutron background due to the PMTs will set a sensitivity limit for the spin-independent WIMP-nucleon cross-section below 10'7 pb, which should be achieved within a few months of deployment underground. An active neutron veto has been planed for retro-fitting around the detector. This hydrocarbon scintillator veto will tag internal neutrons in coincidence with the target. With this system in place, we expect to achieve the design sensitivity of 10"8 pb within a year (an exposure of 2000 kg days). Replacement of the phototubes by new models with even lower radioactivity is being studied, which could improve our WIMP-nucleon sensitivity even further. 4.

Conclusion

ZEPLIN III is in the final stages of commissioning, and will soon undergo the first surface tests. It will then be installed at the Boulby Underground Laboratory. A sensitivity to the spin-independent WIMP-nucleon cross-section as low as 10"8 pb should be achieved, which is well inside the SUSY parameter space where WIMP interactions are more likely to be found.

References 1. T. J. Sumner, Proc. 3rd Int. Workshop on the Identification of Dark Matter, ed. N. J. C. Spooner & V. Kudryavtsev, Singapore: World Scientific, p.452 (2001). T. J. Sumner et al, New Astronomy Reviews (2005) (in press). 2. B. A. Dolgoshein, V. N. Lebedenko & B. U. Rodionov, JETP Lett. 11 (1970)513. 3. D. Cline, Nuc. Phys. B Proc. Sup.) 87 (2000) 114. 4. A. S. Howard et al, Proc. 3rd Int. Workshop on the Identification of Dark Matter, ed. N. J. C. Spooner & V. Kudryavtsev, Singapore: World Scientific, p.457 (2001). 5. D. Yu. Akimov, Proc. 4rd Int. Workshop on the Identification of Dark Matter, ed. N. J. C. Spooner & V. Kudryavtsev, Singapore: World Scientific, p.371 (2003).

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6. H. M. Araujo et al, Nucl. Instrum. & Meth. A521 (2004) 407. 7. H. M. Araujo, ZeplII: A (GEANT4) Simulation Tool for ZEPLIN III, Internal Report, Imperial College London (2001). 8. S. Kubota, Phys. Rev. B17 (1978) 2762. 9. J. Dawson et al, Nucl. Instrum. & Meth. A, in press; physics/0502026. 10. D. Akimov et al, Phys. Lett. B 524 (2002) 245. 11. M. I. Lopes et al, presented at XeSAT2005, Tokyo, Japan, 8-10 March, 2005. 12. E.Aprile et al. (2005), astro-ph/0503621. 13. E. Aprile et al. (2005), astro-ph/0502279. 14. E. M. Gushchin et al, Sov. Phys. JETP 49(5) (1979) 856. 15. D. Davidge, PhD Thesis, Imperial College London (Un. London), 2004. 16. J. Dawson, PhD Thesis, Imperial College London (Un. London), 2003. 17. L. S. Miller et al, Phys. Rev. 166(3) (1968) 871.

D A R K M A T T E R DETECTABILITY W I T H C E R E N K O V TELESCOPES

F. PRADA Institute) de Astrofisica de Andalucia (CSIC) Camino Bajo de Huetor, 50 18008 Granada, Spain E-mail: [email protected] If the dark matter (DM) is made of supersymmetric particles, the central region of the Milky Way should emit gamma rays produced by their annihilation. We are using detailed models of our Galaxy to make accurate predictions of continuum gamma-ray fluxes. We discuss that the most important effect is the compression of the dark matter due to the infall of baryons to the galactic center. Our models predict that the signal could be detected at high confidence levels by imaging atmospheric Cerenkov telescopes assuming that neutralinos make up most of the DM in the Universe.

1. Introduction There is an increasing hope that the new generation of Imaging Atmospheric Cerenkov Telescopes (IACTs) would detect in the very near future the 7rays coming from the annihilation products of the SUSY DM in galaxy halos. The success of such a detection will solve one of the most fundamental questions in Astrophysics and Particle Physics: the nature of the dark matter. The lightest supersymmetric particle (LSP) has been proposed to be a suitable candidate for the non-baryonic cold DM. The LSP is stable in SUSY models where R-parity is conserved and its annihilation cross section and mass leads appropriate relic densities in the range allowed by WMAP, i.e. 0.095 < Q C D M ^ 2 < 0.129. Most of SUSY breaking scenarios yields the lightest neutralino (x?) as the leading candidate for the cold DM. The number of x? annihilations in galaxy halos and therefore the expected 7 signal arriving at the Earth depends not only on the adopted SUSY model but also strongly depends on the DM density p
327

328

area, from a circular aperture on the sky of width crt (the resolution of the telescope) observing at a given direction ^o relative to the center of the MW can be written as: F(E

/SUSY

> Eth)

= ^ / S U S Y • tf (*o),

= ^ ^ . tf(*o) =

(1)

JjmB(Q)dn.

The factor /SUSY/47T represents the isotropic probability of 7-ray production per unit of DM density and depends only on the physics of annihilating Xi particles. It can be determined for any SUSY scenario given the neutralino mass mx, the number of continuum 7-ray photons Ny emitted per annihilation, with energy above the I ACT energy threshold {Eth), and the thermally average cross section (crv). All the astrophysical properties (such as the DM distribution and geometry considerations) appear only in the factor U(^o). This factor also accounts for the beam smearing, where J ( * ) = floa Pdm(r) ^> *s ^he integral of the line-of-sight of the square of the DM density along the direction \I>, and B(Q)d£l is the Gaussian beam of the telescope. A cuspy DM halo p<jm(r) oc r~ Q predicted by the simulations of the Cold Dark Matter with the cosmological constant (ACDM ) is often assumed for the calculations of C/(*o). Cosmological TV-body simulations indicate that the distribution of DM in relaxed halos varies between two shapes: the Navarro, Frenk & White (NFW) density profile with asymptotic slope a = 1 and the steeper Moore et al. profile with slope 1.5. 2. Milky Way mass models with adiabatic compression The predictions for the DM halos are valid only for halos without baryons. When normal gas ("baryons") loses its energy through radiative processes, it falls to the central region of forming galaxy. As the result of this redistribution of mass, the gravitational potential in the center changes substantially. The DM must react to this deeper potential by moving closer to the center and increasing its density. This increase in the DM density is often treated using adiabatic invariants. This is justified because there is a limit to the time-scale of changes in the mass distribution: changes of the potential at a given radius cannot happen faster than the dynamical time-scale denned by the mass inside the radius. Adiabatic contraction of dark matter in a collapsing protogalaxy was used already by Zeldovich et al. (1980) to

329 set constraints of properties of elementary particles (annihilating massive neutrinos). The present form of analytical approximation (circular orbits) was introduced in Blumenthal et al. (1986). If Mm(rin) is the initial distribution of mass (the one predicted by cosmological simulations), then the final (after compression and formation of galaxy) mass distribution is given by Mfjn(r)r = Mjn(r-m)rin, where Mfin = MUM + -Mbar- The approximation assumes that orbits are circular and, thus M(r) is the mass inside the orbit. This is not true for elongated orbits: mass M(r) is smaller than the real mass, which a particle "feels" when it travels along elongated trajectory. This difference in masses requires a relatively small correction: mass M should be replaced by the mass inside time-averaged radius of trajectories passing through given radius r: Mfi n ((r))r = Mi n ((ri n ))ri n . We find the correction using Monte Carlo realizations of trajectories in the NFW equilibrium halo and finding the time-averaged radii (x) w 1.72a; 0,82 /(H-52;) 0 ' 085 , x = r/rs. In order to make realistic predictions for annihilation rates, we construct two detailed models of the MW Galaxy by redoing the full analysis of numerous observational data collected in Klypin et al. 2001. The models are compatible with the available observational data for the MW and their main parameters are given in Table 1 of Prada et al. (2004). Models assume that without cooling the density of baryons is proportional to that of the DM and the final baryon distribution is constrained by the observational data.

3. Computation of

/SUSY

in the m S U G R A scenario

To illustrate the expected variations of /SUSY we focus on the mSUGRA scenario. We are using the Darksusy package (Gondolo et al. 2002) to compute X? relic densities and annihilation properties for 106 mSUGRA random models. We also check that results are not ruled out by present accelerator bounds. We compute a scan of models, which covers the relevant regimes in mSUGRA parameter space. All models within WMAP allowed relic densities leads to \i masses from 70 GeV up to 1400 GeV, approximately. The (av) lies in the range 1-10 -29 to 3-10~26 c m _ 3 s _ 1 . We also compute the / S U S Y / 1 0 - 3 2 dependence with the IACT E t h. For a given E t h, we compute all the mx, {av) and N 7 intervals. This gives the allowed /SUSY region for Xi which can be detected with the IACT, i.e, those with mx>Eth.

330

4. Gamma-ray annihilation observability in the Milky Way The expected x? annihilation 7 flux can be computed from Eq. 1 for the compressed DM density profile provided by our MW models as a function of the angular distance ^0 from the Galactic Center. In Fig. 1 we show the predicted fluxes in units of / S U S Y / 1 0 - 3 2 - We also show as a comparison the expected flux for the uncompressed NFW density profile. The flux profiles were determined for a typical IACT of resolution at = 0.1° (Afi = 10~5sr). We have multiplied the flux profiles by a factor of 1.7 quoted by Stoehr et al. (2003) to account for the presence of substructure inside the MW halo. The success of a detection requires that the minimum detectable 7 flux F m j n for an exposure of t seconds, given an IACT of effective area Aefi, angular resolution at and threshold energy Em exceeds a significant number Ms of standard deviations (Msa) the background noise ^/Nb, i.e. FmmAefit/y/Nb > Ms. The background counts (Nb) due to electronic and hadronic (cosmic protons and helium nucleids) cosmic ray showers have been estimated using the expressions from Bergstrom et al. (1998). As an additional background, one has to consider also the contamination due to isolated muons which depending on the f.o.v. and altitude location of the telescope may be even the dominant background at some energy range. The diffuse galactic and extragalactic gamma radiation are negligible compare to this background. The Eth of an IACT depends on the zenith angle of observation. The Galactic Center is visible at different zenith angles by all present IACTs (e.g. CANGAROO-III, H.E.S.S.,MAGIC, VERITAS), but in the best case an Eth of about 100 GeV can be achieved. The Aeg is also sensible to the zenith angle of observation, here we choose a value of 1 x 109 cm 2 . This detectability condition will allow us to compute the 5
331

NFW DM profile of the Model A will not be detected even in the direction of the Galactic Center. The effect of the adiabatic compression, which was previously ignored, is a crucial factor: in the central ~ 3 kpc of the Milky Way, where the baryons dominate, it does not make sense to use the dark matter profiles provided by cosmological N-body simulations: the DM must fall into the deep potential well created by the collapsed baryons. Thus, the models presented here are not extreme: they are the starting point for realistic predictions of the annihilation fluxes.

I

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Figure 1. Predicted continuum 7 fluxes as a function of distance * o from the Galactic Center. The dashed line give the minimum detectable gamma flux F m i n (see text). Predicted continuum 7 fluxes as a function of distance * o from the Galactic Center. The dashed line give the minimum detectable gamma flux F m j n (see text).

332

References 1. G.R. Blumenthal, S.M. Faber, R. Flores and J.R. Primack, Astrophys. J., 301, 27 (1986). 2. P. Gondolo et al., astro-ph/0211238 (2002). 3. A. Klypin, H. Zhao, and R.S. Somerville, Astrophys. J., 573, 597 (2002). 4. F. Prada, A. Klypin et al. Phys. Rev. Letters, 93, 241301 (2004). 5. F. Stoehr, S.D.M. White, V. Springel, G. Tormen, and N. Yoshida, MNRAS, 345, 1313 (2003). 6. Ya.B. Zeldovich, A.A. Klypin, M.Yu. Khlopov, and V.M. Chechetkin, Soviet J. Nucl. Phys., 31, 664 (1980).

LIST OF PARTICIPANTS Vladan Arsenijevic Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal arsenije@ist. utl.pt

Pedro Abreu LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Sofia Andringa IFAE/UAB Edifici de Ciences 08193 Bellaterra (Barcelona) Spain andringa@ifae. es

Pedro Assis LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Pierre Antilogus LPNHE/IN2P3 4 place Jussieu Tour 33 - Rez de Chaussee 75252 Paris Cedex 05 France P.Antilogus@in2p3. fr

Domingos Barbosa Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal barbosa@astro. up.pt

Henrique Araujo Imperial College London Blackett Laboratory Prince Consort RD London SW7 2BW United Kingdom H.Araujo@imperial. ac. uk

Gaspar Barreira LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Luisa Arruda LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Orfeu Bertolami Instituto Superior Tecnico Departamento de Fisica Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] 333

334

Goncalo Borges LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Ruben Conceicao LIP Av. Elias Garcia 14,1° 1000-149 Lisboa Portugal rconceicao@nfist. ist. utl.pt

Pedro Brogueira Instituto Superior Tecnico Departamento de Fisica Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]. utl.pt

Joao Costa LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Vitor Cardoso McDonnell Center for the Space Sciences Department of Physics Washington University, St. Louis MO 63130 USA [email protected] Nuno Castro LIP Av. Elias Garcia 14, 1° 1000-146 Lisboa Portugal nuno. [email protected] Osvaldo Catalano IASF - Palermo/CNR ViaUgolaMalfa153 90146 Palermo Italy osvaldo. catalano@pa. iasf. cnr. it

Eamonn Daly ESA/ESTEC Keplerlaan 1 Postbus 299 2200 AG Noordwijk The Netherlands eamonn. daly@esa. int Alessandro de Angelis University of Udine Via delle Scienze, 208 1-33100 UDINE Italy deangelis@fisica. uniud. it Jorge Dias de Deus Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

335 Goncalo Dias Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

Sebastien Fabbro Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

Oscar Dias Perimeter Institute for Theoretical Physics 31 Caroline St. N., Waterloo, Ontario N2L 2Y5 Canada odias@perimeterinstitute. ca

Enrique Fernandez IFAE, Edifici Cn. Facultat Ciencies UAB E-08193 Bellaterra (Barcelona) Spain [email protected]

Teresa Dova Departamento de Fisica Universidad Nacional de la Plata CC67 La Plata (1900) Argentina dova@fisica. unlp.edu. ar Dominik Elsaesser University of Wuerzburg Institut Theoretische Physik Astrophysik Am Hubland, D-97074 Wuerzburg Germany [email protected] Catarina Espirito Santo LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Manuel Fiolhais Universidade de Coimbra Departamento de Fisica 3004-516 Coimbra Portugal tmanuel@teor. fis. uc.pt Margarida Fraga LIP-Coimbra Departamento de Fisica - Univ. Coimbra Rua Larga, 3004-516 Coimbra Portugal [email protected]. uc.pt Sijie Gao Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

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Nicola Giglietto INFN - Bari Via E. Orabona, 4 1-70126 Bari Italy giglietto@ba. infn. it Patricia Goncalves LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Alfredo Barbosa Henriques Institute) Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal alfredo@fisica. ist. utl.pt Antonio Insolia University of Catania Dipartimento di Fisica e Astronomia Via. S. Sofia, 64 95123 Catania Italy Antonio. lnsolia@ct. infn. it Charles Jui University of Utah 115 S. 1400 E. #201 JFB Salt Lake City, Utah 84112 USA jui@cosmic. utah. edu

Kostas Kokkotas Aristotle University Department of Physics Section of Astrophysics, Astronomy and Mechanics 541 24 Thessaloniki Macedonia Greece kokkotas@auth. gr Ralf Lehnert University of Vanderbilt Department of Physics and Astronomy 6301 Stevenson Center Nashville, Tennessee 37235 USA [email protected] Jose Sande Lemos Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal lemos@fisica. ist. utl.pt Cipriano Lomba EFACEC Sistemas de Electronica, S.A. Rua Eng. Frederico Ulrich, Apartado 3081 4471-907 Moreira - Maia Portugal [email protected]

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llidio Lopes Instituto Superior Tecnico CENTRA & Universidade de Evora Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

Joao Magueijo Theoretical Physics Group Department of Physics Imperial College Prince Consort RD., London SW7 2BZ United Kingdom j. magueijo@imperial. ac. uk Dalmiro Maia Universidade do Porto Faculdade de Ciencias Observatorio Astronomico Alameda do Monte da Virgem 4130-146 Vila Nova de Gaia Portugal [email protected]

Ana Maria Mourao Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal amourao@ist. utl.pt Sergio Navas Concha University of Granada Departamento de Fisica Teorica y del Cosmos Av. Severo Ochoa s/n 18071 Granada Spain [email protected] Rui Neves Universidade do Algarve Fac. de Ciencias e Tecnologia Departamento de Fisica e CENTRA Campus de Gambelas 8005-139 Faro Portugal [email protected]

Amelia Maio LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Antonio Onofre LIP-Coimbra Departamento de Fisica - Univ. Coimbra Rua Larga, 3004-516 Coimbra Portugal antonio.onofre@cern. ch

Jose Maneira LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

Catarina Ortigao LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

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Jorge Paramos Instituto Superior Tecnico Departamento de Fisica Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected] Miguel Paulos LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Rui Pereira LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Tiago Pereira Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal tiago.pereira@ist. utl.pt Mario Pimenta LIP Av. Elias Garcia 14,1° 1000-149 Lisboa Portugal [email protected]

Robertus Potting Universidade do Algarve Fac. de Ciencias e Tecnologia Departamento de Fisica e CENTRA Campus de Gambelas 8005-139 Faro Portugal [email protected] Francisco Prada Instituto de Astrofisica de Andalucia Camino Bajo de Huetor, 50 18008 Granada Spain [email protected] Pedro Rato Universidade do Algarve Fac. de Ciencias e Tecnologia LIP Campus de Gambelas 8005-139 Faro Portugal [email protected] Luciano Rezzolla SISSA/ISAS Via Beirut 2-4 34014 Trieste Italy [email protected] Pedro Rodrigues LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected]

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Jorge Romao Institute Superior Tecnico CFTP Av. Rovisco Pais 1049-001 Lisboa Portugal jorge. romao@ist. utl.pt Paulo M. Sa Universidade do Algarve Fac. de Ciencias e Tecnologia Departamento de Fisica e CENTRA Campus de Gambelas 8005-139 Faro Portugal [email protected] Nuno Santos Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal nlsantos@fisica. ist. utl.pt Rui Santos LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Fabio Scardigli Instituto Superior Tecnico CENTRA Av. Rovisco Pais 1049-001 Lisboa Portugal [email protected]

Peter Sonderegger LIP/CERN Av. Elias Garcia 14,1° 1000-149 Lisboa Portugal [email protected] Andreia Trindade LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Bernardo Tome LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal [email protected] Brigitte Tome Universidade do Algarve Fac. de Ciencias e Tecnologia CENTRA Campus de Gambelas 8005-139 Faro Portugal [email protected] Filipe Veloso LIP Av. Elias Garcia 14, 1° 1000-149 Lisboa Portugal filipe. [email protected]

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Pedro Viana CAUP Rua das Estrelas, s/n 4150-762 Porto Portugal viana@astro. up.pt

Alan Watson School of Physics and Astronomy University of Leeds Leeds LS2 9JT United Kingdom a. a. watson@leeds. ac. uk

In international workshops on "New Worlds in Astroparticle Physics" held biannually, astronomers, astrophysicists and particle physicists discuss recent developments in the exciting and rapidly developing field of Astroparticle Physics. Similar to previous workshops, this 5th international workshop introduced

NEW WORLDS IN

experimental, observational and theoretical subjects through review lectures. This was followed by shorter contributions on the recent developments in Astroparticle Physics. This workshop covered an array of subjects like cosmic rays, gravitational waves, space radiation, neutrino physics, cosmological parameters, black

Proceedings ol the Fifth Interrational Workshop

holes, dark matter and dark energy.

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